WORKS  OF 

GEORGE  L.  HOSMER 

PUBLISHED    BY 

JOHN  WILEY  &   SONS,   INC. 


Text-Book  on  Practical  Astronomy 

Second  edition,  214  pages,  6  by  9,  78  cuts,  cloth, 
$2.00  net. 

Azimuth 

Second  edition,  78  pages,  45  by  7,  6  cuts,  flexible 
binding,  $1.00  net. 

Geodesy 

First  edition,  377  pages,  6  by  9,  115  cuts,  cloth, 
$3.50  net. 

Navigation 

First  edition,  214  pages,  4^  by  6|,  52  cuts,  cloth, 
$1.25  net. 

BY 
CHARLES  B.   BREED 

AND 

GEORGE  L.  HOSMER 

The  Principles  and  Practice  of  Surveying 

Vol.  I,  Elementary  Surveying,  Fourth  edition, 
610  pages,  5  by  7$,  216  cuts,  flexible  binding, 
pocket  size,  $3.00. 

Vol.  II,  Higher  Surveying,  Second  edition,  460 
pages,  5  by  7^,  158  cuts,  flexible  binding,  pocket 
size,  $2.50. 


TEXT-BOOK 

ON 


PRACTICAL   ASTRONOMY 


BY 

GEORGE    L.  HOSMER 

Associate  Professor  of  Topographical  Engineering,  Massachusetts 
Institute  of  Technology 


SECOND  EDITION,   REVISED 


NEW  YORK 

JOHN    WILEY   &    SONS,   INC. 
LONDON:  CHAPMAN  &  HALL,  LIMITED 


COPYRIGHT,  1910  AND  1917 

BY 
GEORGE    L.    HOSMER 


••"••»•  ••••••*•*•• 


Stanhope  ipress 

F.    H.   GILSON     COMPANY 
BOSTON,     U.S.A. 


H7 


PREFACE 


THE  purpose  of  this  volume  is  to  furnish  a  text  in  Practical 
Astronomy  especially  adapted  to  the  needs  of  civil-engineering 
students  who  can  devote  but  little  time  to  the  subject,  and  who 
are  not  likely  to  take  up  advanced  study  of  Astronomy.  The 
text  deals  chiefly  with  the  class  of  observations  which  can  be 
made  with  surveying  instruments,  the  methods  applicable  to 
astronomical  and  geodetic  instruments  being  treated  but  briefly. 
It  has  been  the  author's  intention  to  produce  a  book  which  is 
intermediate  between  the  text-book  written  for  the  student  of 
Astronomy  or  Geodesy  and  the  short  chapter  on  the  subject 
generally  given  in  text-books  on  Surveying.  The  subject  has 
therefore  been  treated  from  the  standpoint  of  the  engineer,  who 
is  interested  chiefly  in  obtaining  results,  and  those  refinements 
have  been  omitted  which  are  beyond  the  requirements  of  the 
work  which  can  be  performed  with  the  engineer's  transit.  This 
has  led  to  the  introduction  of  some  rather  crude  mathematical 
processes,  but  it  is  hoped  that  these  are  presented  in  such  a  way 
as  to  aid  the  student  in  gaining  a  clearer  conception  of  the  prob- 
lem without  conveying  wrong  notions  as  to  when  such  short-cut 
methods  can  properly  be  applied.  The  elementary  principles 
have  been  treated  rather  elaborately  but  with  a  view  to  making 
these  principles  clear  rather  than  to  the  introduction  of  refine- 
ments. Much  space  has  been  devoted  to  the  Measurement  of 
Time  because  this  subject  seems  to  cause  the  student  more 
difficulty  than  any  other  branch  of  Practical  Astronomy.  The 
attempt  has  been  made  to  arrange  the  text  so  that  it  will  be  a 
convenient  reference  book  for  the  engineer  who  is  doing  field 
work. 

For  convenience  in  arranging  a  shorter  course  those  subjects 


425853 


iv  PREFACE 

which  are  most  elementary  are  printed  in  large  type.  The  mat- 
ter printed  in  smaller  type  may  be  included  in  a  longer  course 
and  will  be  found  convenient  for  reference  in  field  practice,  par- 
ticularly that  contained  in  Chapters  X  to  XIII. 

The  author  desires  to  acknowledge  his  indebtedness  to  those 
who  have  assisted  in  the  preparation  of  this  book,  especially  to 
Professor  A.  G.  Robbins  and  Mr.  J.  W.  Howard  of  the  Massa- 
chusetts Institute  of  Technology  and  to  Mr.  F.  C.  Starr  of  the 
George  Washington  University  for  valuable  suggestions  and  crit- 
icisms of  the  manuscript. 

G.  L.  H. 

BOSTON,  June,  1910. 


PREFACE    TO    THE    SECOND    EDITION 


RECENT  changes  in  the  form  of  the  tables  of  the  Nautical 
Almanac  have  made  it  necessary  to  rewrite  Chapter  VI.  Ad- 
vantage has  been  taken  of  this  opportunity  to  make  certain 
other  desirable  changes  in  the  book.  The  most  important 
additions  are  (i)  the  introduction  of  special  tables  for  comput- 
ing the  azimuth  of  Polaris  at  any  hour  angle,  (2)  the  addition 
of  formulae  [21]  and  [27],  which  are  often  more  convenient  for 
the  engineer  than  the  forms  previously  given,  and  (3)  a  general 
revision  of  the  star  maps.  Table  V  has  been  replaced  by  a 
more  complete  one,  and  numerous  minor  changes  have  been 
made  in  the  cuts  and  the  numerical  examples. 

G.  L.  H. 

BOSTON,  MASS.,  May,  1916. 


TABLE  OF  CONTENTS 


CHAPTER  I 
THE  CELESTIAL  SPHERE  —  REAL  AND  APPARENT  MOTIONS 

ART.  PAGE 

1.  Practical  Astronomy i 

2.  The  Celestial  Sphere i 

3.  Apparent  Motion  of  the  Sphere 3 

4.  The  Motions  of  the  Planets 3 

5.  Meaning  of  Terms  East  and  West 6 

6.  The  Earth's  Orbital  Motion  —  The  Seasons 7 

7.  The  Sun's  Apparent  Position  at  Different  Seasons 9 

8.  Precession  and  Nutation 10 

9.  Aberration  of  Light 12 


CHAPTER  II 
DEFINITIONS  —  POINTS  AND  CIRCLES  OF  REFERENCE 

10.  Definitions 14 

Vertical  Line  —  Zenith  —  Nadir  —  Horizon  —  Vertical  Circles 
—  Almucantars  —  Poles  —  Equator  —  Hour  Circles  —  Par- 
allels of  Declination  —  Meridian  —  Prime  Vertical  —  Eclip- 
tic —  Equinoxes  —  Solstices  —  Colures. 


CHAPTER  III 

SYSTEMS  OF  COORDINATES  ON  THE  SPHERE 

11.  Spherical  Coordinates 18 

12.  The  Horizon  System 19 

13.  The  Equator  Systems 19 

15.  Coordinates  of  the  Observer 22 

16.  Relation  between  the  Two  Systems  of  Coordinates 23 


vi  TABLE  OF  CONTENTS 

CHAPTER  IV 
RELATION  BETWEEN  COORDINATES 

ART.  PAGE 

17.  Relation  between  Altitude  of  Pole  and  Latitude  of  Observer.  ...       27 

18.  Relation  between  Latitude  of  Observer  and  the  Declination  and 

Altitude  of  a  Star  on  the  Meridian 30 

19.  The  Astronomical  Triangle 31 

20.  Relation  between  Right  Ascension  and  Hour  Angle 36 


CHAPTER  V 
MEASUREMENT  or  TIME 

21.  The  Earth's  Rotation 39 

22.  Transit  or  Culmination 39 

23.  Sidereal  Day 39 

24.  Sidereal  Time 40 

25.  Solar  Day 40 

26.  Solar  Time 40 

27.  Equation  of  Time 41 

28.  Conversion  of  Apparent  Time  into  Mean  Time  and  vice  versa  ...  43 

29.  Astronomical  and  Civil  Time 44 

30.  Relation  between  Longitude  and  Time 45 

31.  Relation  between  Sidereal  Time,  Right  Ascension  and  Hour  Angle 

of  any  Point  at  a  Given  Instant 48 

32.  Star  on  the  Meridian 49 

33.  Relation  between  Mean  Solar  and  Sidereal  Intervals  of  Time 49 

34.  Relation  between  Sidereal  and  Mean  Time  at  any  Instant 52 

35.  Standard  Time 56 

36.  The  Date  Line 58 

37.  The  Calendar 59 


CHAPTER  VI 

THE  AMERICAN  EPHEMERIS  AND  NAUTICAL  ALMANAC  —  STAR 
CATALOGUES  —  INTERPOLATION 

38.  The  Ephemeris 62 

39.  Star  Catalogues 65 

40.  Interpolation 68 


TABLE   OF   CONTENTS  v11 

CHAPTER  VII 
THE  EARTH'S  FIGURE  —  CORRECTIONS  TO  OBSERVED  ALTITUDES 

ART.  PAGE 

41.  The  Earth's  Figure 72 

42.  Parallax 73 

43.  Refraction 76 

44.  Semidiameters 78 

45-   Dip 79 

46.  Sequence  of  Corrections 80 

CHAPTER  VIII 

DESCRIPTION  OF  INSTRUMENTS  —  OBSERVING 

47.  The  Engineer's  Transit 82 

48.  Elimination  of  Errors 83 

49.  Attachments  to  the  Engineer's  Transit  —  Reflector 86 

50.  Prismatic  Eyepiece 87 

51.  Sun  Glass 87 

52.  The  Portable  Astronomical  Transit 87 

53.  The  Sextant 88 

54.  Artificial  Horizon 91 

55.  Chronometer 92 

56.  Chronograph 93 

57.  The  Zenith  Telescope 94 

58.  Suggestions  about  Observing 95 

CHAPTER  IX 

THE  CONSTELLATIONS 

59.  The  Constellations 98 

60.  Method  of  Naming  Stars 98 

61.  Magnitudes 99 

62.  Constellations  near  the  Pole 99 

63.  Constellations  near  the  Equator 100 

64.  The  Planets 102 

CHAPTER  X 
.  »  OBSERVATIONS  FOR  LATITUDE 

65.  Latitude  by  a  Circumpolar  Star  at  Culmination 103 

66.  Latitude  by  Altitude  of  the  Sun  at  Noon 105 

67.  Latitude  by  the  Meridian  Altitude  of  a  Southern  Star 107 


Vlll  TABLE   OF   CONTENTS 

ART.  PAGE 

68.  Latitude  by  Altitudes  Near  the  Meridian 108 

69.  Latitude  by  Polaris  when  the  Time  is  Known no 

70.  Precise  Latitude  Determinations  —  Talcott's  Method 112 

CHAPTER  XI 

OBSERVATIONS  FOR  DETERMINING  THE  TIME 

71.  Observations  for  Local  Time 114 

72.  Time  by  Transit  of  a  Star 114 

73.  Observations  with  Astronomical  Transit 117 

74.  Selecting  Stars  for  Transit  Observations 117 

75.  Time  by  Transit  of  the  Sun 119 

76.  Time  by  Altitude  of  the  Sun 1 20 

77.  Time  by  Altitude  of  a  Star 123 

78.  Time  by  Transit  of  Star  over  Vertical  Circle  through  Polaris. ...  124 

79.  Time  by  Equal  Altitudes  of  a  Star 127 

80.  Time  by  Two  Stars  at  Equal  Altitudes 128 

83.  Rating  a  Watch  by  Transit  of  a  Star  over  a  Range 135 

84.  Time  Service 136 

CHAPTER  XII 

OBSERVATIONS  FOR  LONGITUDE 

85.  Methods  of  Measuring  Longitude 139 

86.  Longitude  by  Transportation  of  Timepiece 139 

87.  Longitude  by  the  Electric  Telegraph 140 

88.  Longitude  by  Transit  of  the  Moon 141 

CHAPTER  XIII 

OBSERVATIONS  FOR  AZIMUTH 

89.  Determination  of  Azimuth 146 

90.  Azimuth  Mark 146 

91.  Azimuth  by  Polaris  at  Elongation 147 

92.  Observations  near  Elongation 149 

93.  Azimuth  by  an  Altitude  of  the  Sun 151 

94.  Azimuth  by  an  Altitude  of  a  Star 155 

95.  Azimuth  Observation  on  a  Circumpolar  Star  at  any  Hour  Angle .  155 

96.  The  Curvature  Correction 158 


TABLE  OF   CONTENTS  IX 

ART.  PAGE 

97.  The  Level  Correction 158 

98.  Diurnal  Aberration 158 

99.  Meridian  by  Polaris  at  Culmination 161 

TOO.  Azimuth  by  Equal  Altitudes  of  a  Star 164. 

101.  Observation  for  Meridian  by  Equal  Altitudes  of  the  Sun 165 

102.  Observation  of  the  Sun  near  Noon 166 

103.  Approximate  Azimuth  of  Polaris  when  the  Time  is  Known 167 

CHAPTER  XIV 

NAUTICAL  ASTRONOMY 

104.  Observations  at  Sea 170 

Determination  of  Latitude  at  Sea: 

105.  Latitude  by  Noon  Altitude  of  the  Sun 170 

106.  Latitude  by  Ex-Meridian  Altitudes .... . %. 171 

Determination  of  Longitude  at  Sea: 

107.  Longitude  by  the  Greenwich  Time  and  the  Sun's  Altitude 172 

108.  Longitude  by  the  Lunar  Distance 172 

109.  Azimuth  of  the  Sun  at  a  Given  Time .  174 

no.   Azimuth  of  the  Sun  by  Altitude  and  Time 175 

in.   Sumner's  Method  of  Determining  a  Ship's  Position 175 

112.   Position  by  Computation 178 


TABLES 

I.    MEAN  REFRACTION 184 

II.    CONVERSION  OF  SIDEREAL  INTO  SOLAR  TIME 185 

III.  CONVERSION  OF   SOLAR  INTO  SIDEREAL  TIME 186 

IV.  (A)    SUN'S  PARALLAX  —  (B)    SUN'S   SEMIDIAMETER  —  (C)    DIP   OF 

HORIZON 187 

V.    TIMES  OF  CULMINATION  AND  ELONGATION  OF  POLARIS 188 

VI.    CORRECTION  TO  OBSERVED  ALTITUDE  OF  POLARIS 189 

VII.    VALUES  OF  FACTOR  112.5  X  3600  X  sin  i"  X  tan  Ze 190 

GREEK  ALPHABET 190 

LIST  OF  ABBREVIATIONS 19* 


PRACTICAL  ASTRONOMY 


CHAPTER  I 

THE  CELESTIAL   SPHERE  — REAL  AND  APPARENT 

MOTIONS 

1.  Practical  Astronomy. 

Practical  Astronomy  treats  of  the  theory  and  use  of  astro- 
nomical instruments  and  the  methods  of  computing  the  results 
obtained  by  observation.  The  part  of  the  subject  which  is  of 
especial  importance  to  the  surveyor  is  that  which  deals  with  the 
methods  of  locating  points  on  the  earth's  surface  and  of  ori- 
enting the  lines  of  a  survey,  and  includes  the  determination  of 
(i)  latitude,  (2)  time,  (3)  longitude,  and  (4)  azimuth.  In  solving 
these  problems  the  observer  makes  measurements  of  the  direc- 
tions of  the  sun,  moon,  stars,  and  other  heavenly  bodies;  he  is 
not  concerned  with  the  distances  of  these  objects,  with  their 
actual  motions  in  space,  nor  with  their  physical  characteristics, 
but  simply  regards  them  as  a  number  of  visible  objects  of  known 
positions  from  which  he  can  make  his  measurements. 

2.  The  Celestial  Sphere. 

Since  it  is  only  the  directions  of  these  objects  that  are  required 
in  practical  astronomy,  it  is  found  convenient  to  regard  all 
heavenly  bodies  as  being  situated  on  the  surface  of  a  sphere 
whose  radius  is  infinite  and  whose  centre  is  at  the  eye  of  the 
observer.  The  apparent  position  of  any  object  on  the  sphere  is 
found  by  imagining  a  line  drawn  from  the  eye  to  the  object,  and 
prolonging  it  until  it  pierces  the  sphere.  For  example,  the 
apparent  position  of  Si  on  the  sphere  (Fig.  i)  is  at  5/,  which  is 
supposed  to  be  at  an  infinite  distance  from  C;  the  position  of 
$2  is  S2',  etc.  By  means  of  this  imaginary  sphere  all  problems 


PRACTICAL  ASTRONOMY 


involving  the  angular  distances  between  points,  and  angles 
between  planes  through  the  centre  of  the  sphere,  may  readily  be 
solved  by  applying  the  formulae  of  spherical  trigonometry. 
This  device  is  not  only  convenient  for  mathematical  purposes, 
but  it  is  perfectly  consistent  with  what  we  see,  because  all  celestial 
objects  are  so  far  away  that  they  appear  to  the  eye  to  be  at  the 
same  distance,  and  consequently  on  the  surface  of  a  great  sphere. 


FIG.  i.    APPARENT  POSITIONS  ON  THE  SPHERE 

From  the  definition  it  will  be  apparent  that  each  observer  sees 
a  different  celestial  sphere,  but  this  causes  no  actual  inconve- 
nience, for  distances  between  points  on  the  earth's  surface  are  so 
short  when  compared  with  astronomical  distances  that  they  are 
practically  zero  except  for  the  nearer  bodies  in  the  solar  system. 
This  may  be  better  understood  from  the  statement  that  if  the 
entire  solar  system  be  represented  as  occupying  a  field  one  mile 
in  diameter  the  nearest  star  would  be  about  5000  miles  away  on 
the  same  scale;  furthermore  the  earth's  diameter  is  but  a  minute 
fraction  of  the  distance  across  the  solar  system,  the  ratio  being 
about  8000  miles  to  5,600,000,000  miles,*  or  one  700,000 th  part 
of  this  distance. 


*  The  diameter  of  Neptune's  orbit. 


THE  CELESTIAL  SPHERE  3 

Since  the  radius  of  the  celestial  sphere  is  infinite,  all  of  the 
lines  in  a  system  of  parallels  will  pierce  the  sphere  in  the  same 
point,  and  parallel  planes  at  any  finite  distance  apart  will  cut 
the  sphere  in  the  same  great  circle.  This  must  be  kept  constantly 
in  mind  when  representing  the  sphere  by  means  of  a  sketch,  in 
which  minute  errors  will  necessarily  appear  to  be  very  large. 
The  student  should  become  accustomed  to  thinking  of  the 
appearance  of  the  sphere  both  from  the  inside  and  from  an  out- 
side point  of  view.  It  is  usually  easier  to  understand  the  spheri- 
cal problems  by  studying  a  small  globe,  but  when  celestial 
objects  are  actually  observed  they  are  necessarily  seen  from  a 
point  inside  the  sphere. 

3.  Apparent  Motion  of  the  Celestial  Sphere. 

If  a  person  watches  the  stars  for  several  hours  he  will  see  that 
they  appear  to  rise  in  the  east  and  to  set  in  the  west,  and  that 
their  paths  are  arcs  of  circles.  By  facing  to  the  north  (in  the 
northern  hemisphere)  it  will  be  found  that  the  circles  are  smaller 
and  all  appear  to  be  concentric  about  a  certain  point  in  the  sky 
called  the  pole ;  if  a  star  were  exactly  at  this  point  it  would  have 
no  apparent  motion.  In  other  words,  the  whole  celestial  sphere 
appears  to  be  rotating  about  an  axis.  This  apparent  rotation 
is  found  to  be  due  simply  to  the  actual  rotation  of  the  earth 
about  its  axis  (from  west  to  east)  in  the  opposite  direction  to 
that  in  which  the  stars  appear  to  move.* 

4.  Motions  of  the  Planets. 

If  an  observer  were  to  view  the  solar  system  from  a  point  far 
outside,  looking  from  the  north  toward  the  south,  he  would  see 
that  all  of  the  planets  (including  the  earth)  revolve  about  the 
sun  in  elliptical  orbits  which  are  nearly  circular,  the  direction 
of  the  motion  being  counter-clockwise  or  left-handed  rotation. 

*  This  apparent  rotation  may  be  easily  demonstrated  by  taking  a  photo- 
graph of  the  stars  near  the  pole,  exposing  the  plate  for  several  hours.  The 
result  is  a  series  of  concentric  arcs  all  subtending  the  same  angle.  If  the 
camera  is  pointed  southward  and  high  enough  to  photograph  stars  near  the 
equator  the  star  trails  appear  as  straight  lines. 


PRACTICAL  ASTRONOMY 


He  would  also  see  that  the  earth  rotates  on  its  axis,  once 
per  day,  in  a  counter-clockwise  direction.  The  moon  revolves 
around  the  earth  in  an  orbit  which  is  not  so  nearly  circular, 
but  the  motion  is  in  the  same  (left-handed)  direction.  The 


FIG.  2.    DIAGRAM  OF  THE  SOLAR  SYSTEM  WITHIN  THE  ORBIT  OF  SATURN 

apparent  motions  resulting  from  these  actual  motions  are  as 
follows:  The  whole  celestial  sphere,  carrying  with  it  all  the 
stars,  sun,  moon,  and  planets,  appears  to  rotate  about  the  earth's 
axis  once  per  day  in  a  clockwise  (right-handed)  direction.  The 
stars  change  their  positions  so  slowly  that  they  appear  to  be  fixed 
in  position  on  the  sphere,  whereas  all  objects  within  the  solar 
system  rapidly  change  their  apparent  positions  among  the  stars. 
For  this  reason  the  stars  are  called  fixed  stars  to  distinguish  them 
from  the  planets;  the  latter,  while  closely  resembling  the  stars 


THE  CELESTIAL  SPHERE  5 

in  appearance,  are  really  of  an  entirely  different  character.  The 
sun  appears  to  move  slowly  eastward  among  the  stars  at  the  rate 
of  about  i°  per  day,  and  to  make  one  revolution  around  the  earth 


FIG.  3a.    SUN'S  APPARENT  POSITION  AT  GREENWICH  NOON  ON  MAY  22,  23, 

AND  24,  1910 


,10° 

Y  iv  HI 

FIG.  3b.    MOON'S  APPARENT  POSITION  AT  14^  ON  FEB.  15,  16,  AND  17,  1910 

in  just  one  year.  The  moon  also  travels  eastward  among  the 
stars,  but  at  a  much  faster  rate;  it  moves  an  amount  equal  to 
its  own  diameter  in  about  an  hour,  and  completes  one  revolu- 


PRACTICAL  ASTRONOMY 


tion  in  a  lunar  month.  Figs.  3 a  and  3b  show  the  daily  motions 
of  the  sun  and  moon  respectively,  as  indicated  by  their  plotted 
positions  when  passing  through  the  constellation  Taurus.  It 
should  be  observed  that  the  motion  of  the  moon  eastward  among 
the  stars  is  an  actual  motion,  not  merely  an  apparent  one  like 
that  of  the  sun.  The  planets  all  move  eastward  among  the 
stars,  but  since  we  ourselves  are  on  a  moving  object  the  motion 
we  see  is  a  combination  of  the  real  motions  of  the  planets  around 

+  5° 


VIRGO 


-5'- 


-10 


-15C 


Spica, 


XIII  XII 

FIG.  4.    APPARENT  PATH  OF  JUPITER  FROM  Oca:.,  1909  TO  OCT.,  1910. 

the  sun  and  an  apparent  motion  caused  by  the  earth's  revolution 
around  the  sun;  the  planets  consequently  appear  at  certain 
times  to  move  westward  (i.e.,  backward),  or  to  retrograde. 
Fig.  4  shows  the  loop  in  the  apparent  path  of  the  planet  Jupiter 
caused  by  the  earth's  motion  around  the  sun.  It  will  be  seen 
that  the  apparent  motion  of  the  planet  was  direct  except  from 
January  to  June,  1910,  when  it  had  a  retrograde  motion. 

5.   Meaning  of  Terms  East  and  West. 

In  astronomy  the  terms  "  east  "  and  "  west  "  cannot  be  taken 
to  mean  the  same  as  they  do  when  dealing  with  directions  in  one 


THE   CELESTIAL   SPHERE 


plane.  In  plane  surveying  "  east  "  and  "  west  "  may  be, con- 
sidered to  mean  the  directions  perpendicular  to  the  meridian 
plane.  If  a  person  at  Greenwich 
(England)  and  another  person  at 
the  1 80°  meridian  should  both 
point  due  east,  they  would  actu- 
ally be  pointing  to  opposite  points 
of  the  sky.  In  Fig.  5  all  four  of 
the  arrows  are  pointing  east  at  the 
places  shown.  It  will  be  seen  from 
this  figure  that  the  terms  "  east  " 
and  "  west "  must  therefore  be 
taken  to  mean  directions  of  ro- 
tation. 

6.   The  Earth's  Orbital  Motion.  —  The  Seasons. 

The  earth  moves  eastward  around  the  sun  once  a  year  in  an 
orbit  which  lies  (very  nearly)  in  one  plane  and  whose  form  is  that 


FIG.  5.    ARROWS  ALL  POINT 
EASTWARD 


b      b 
FIG.  6.    THE  EARTH'S  ORBITAL  MOTION 

of  an  ellipse,  the  sun  being  at  one  of  the  foci.  Since  the  earth  is 
maintained  in  its  position  by  the  force  of  gravitation,  it  moves,  as 
a  consequence,  at  such  a  speed  in  each  part  of  its  path  that  the 


8 


PRACTICAL  ASTRONOMY 


line  joining  the  earth  and  sun  moves  over  equal  areas  in  equal 
times.  In  Fig.  6  all  of  the  shaded  areas  are  equal  and  the  arcs 
aa',  W ,  cc'  represent  the  distances  passed  over  in  the  same  num- 
ber of  days.* 

The  axis  of  rotation  of  the  earth  is  inclined  to  the  plane  of  the 
orbit  at  an  angle  of  about  66°|,  that  is,  the  plane  of  the  earth's 
equator  is  inclined  at  an  angle  of  about  23°^  to  the  plane  of  the 
orbit.  This  latter  angle  is  known  as  the  obliquity  of  the  ecliptic. 
(See  Chapter  II.)  The  direction  of  the  earth's  axis  of  rotation 
is  nearly  constant  and  it  therefore  points  nearly  to  the  same 
place  in  the  sky  year  after  year. 

The  changes  in  the  seasons  are  a  direct  result  of  the  inclination 
of  the  axis  and  of  the  fact  that  the  axis  remains  nearly  parallel 

Vernal  Equinox 
(March  21) 


Summer  Solstice 
(June  21) 


Aphelion 


Perihelion  (Dec.  31) 


Winter  Solstice 
(Dec.  21) 


Autumnal  Equinox 
(Sept.  22) 

FIG.  7.    THE  SEASONS 

to  itself.  When  the  earth  is  in  that  part  of  the  orbit  where  the 
northern  end  of  the  axis  is  pointed  away  from  the  sun  (Fig.  7) 
it  is  winter  in  the  northern  hemisphere.  The  sun  appears  to  be 

*  The  eccentricity  of  the  ellipse  shown  in  Fig.  6  is  exaggerated  for  the  sake 
of  clearness  ;  the  earth's  orbit  is  in  reality  much  more  nearly  circular,  the 
variation  in  the  earth's  distance  from  the  sun  being  only  about  three  per  cent. 


THE   CELESTIAL    SPHERE  9 

farthest  south  about  Dec.  21,  and  at  this  time  the  days  are 
shortest  and  the  nights  are  longest.  When  the  earth  is  in  this 
position,  a  plane  through  the  axis  and  perpendicular  to  the  plane 
of  the  orbit  will  pass  through  the  sun.  About  ten  days  later  the 
earth  passes  the  end  of  the  major  axis  of  the  ellipse  and  is  at  its 
point  of  nearest  approach  to  the  sun,  or  perihelion.  Although 
the  earth  is  really  nearer  to  the  sun  in  winter  than  in  summer, 
this  has  but  a  small  effect  upon  the  seasons;  the  chief  reasons 
why  it  is  colder  in  winter  are  that  the  day  is  shorter  and  the 
rays  of  sunlight  strike  the  surface  of  the  ground  more  obliquely. 
The  sun  appears  to  be  farthest  north  about  June  22,  at  which 
time  summer  begins  in  the  northern  hemisphere  and  the  days  are 
longest  and  the  nights  shortest.  When  the  earth  passes  the 
other  end  of  the  major  axis  of  the  ellipse  it  is  farthest  from  the 
sun,  or  at  aphelion.  On  March  21  the  sun  is  in  the  plane  of 
the  earth's  equator  and  day  and  night  are  of  equal  length  at  all 
places  on  the  earth  (Fig.  7).  On  Sept.  22  the  sun  is  again  in 
the  plane  of  the  equator  and  day  and  night  are  everywhere 
equal.  These  two  times  are  called  the  equinoxes  (vernal  and 
autumnal),  and  the  points  in  the  sky  where  the  sun's  centre  ap- 
pears to  be  at  these  two  dates  are  called  the  equinoctial  points, 
or  more  commonly  the  equinoxes. 

7.   The  Sun's  Apparent  Position  at  Different  Seasons. 

The  apparent  positions  of  the  sun  on  the  celestial  sphere 
corresponding  to  these  different  positions  of  the  earth  are  shown 
in  Fig.  8.  As  a  result  of  the  sun's  apparent  eastward  motion 
from  day  to  day  along  a  path  which  is  inclined  to  the  equator, 
the  angular  distance  of  the  sun  from  the  equator  is  continually 
changing.  Half  of  the  year  it  is  north  of  the  equator  and  half  of 
the  year  it  is  south.  On  June  22  the  sun  is  in  its  most  northerly 
position  and  is  visible  more  than  half  the  day  to  a  person  in  the 
northern  hemisphere  (/,  Fig.  8).  On  Dec.  21  it  is  farthest  south 
of  the  equator  and  is  visible  less  than  half  the  day  (Z>,  Fig.  8). 
In  between  these  two  extremes  it  moves  back  and  forth  across 
the  equator,  passing  it  about  March  21  and  Sept.  22  each  year. 


10  PRACTICAL  ASTRONOMY 

The  apparent  motion  of  the  sun  is  therefore  a  helical  motion 
about  the  axis,  that  is,  the  sun,  instead  of  following  the  path 
which  would  be  followed  by  a  fixed  star,  gradually  increases  or 
decreases  its  angular  distance  from  the  pole  at  the  same  time 
that  it  revolves  once  a  day  around  the  earth.  The  sun's  motion 
eastward  on  the  celestial  sphere,  due  to  the  earth's  orbital  motion, 


E 
FIG.  8.    SUN'S  APPARENT  POSITION  AT  DIFFERENT  SEASONS 

is  not  noticed  until  the  sun's  position  is  carefully  observed  with 
reference  to  the  stars.  If  a  record  is  kept  for  a  year  showing 
which  constellations  are  visible  in  the  east  soon  after  sunset, 
it  will  be  found  that  these  change  from  month  to  month,  and  at 
the  end  of  a  year  the  one  first  seen  will  again  appear  in  the  east, 
showing  that  the  sun  has  apparently  made  the  circuit  of  the 
heavens  in  an  eastward  direction 

8.  Precession  and  Nutation. 

While  the  direction  of  the  earth's  rotation  axis  is  so  nearly 
constant  that  no  change  is  observed  during  short  periods  of 
time,  there  is  in  reality  a  very  slow  progressive  change  in  its 
direction.  This  change  is  due  to  the  fact  that  the  earth  is  not 
quite  spherical  in  form  but  is  spheroidal,  and  there  is  in  conse- 
quence a  ring  of  matter  around  the  equator  upon  which  the 
sun  and  the  moon  exert  a  force  of  attraction  which  tends  to  pull 
the  plane  of  the  equator  into  coincidence  with  the  plane  of  the 
orbit.  But  since  the  earth  is  rotating  with  a  high  velocity  and 


THE   CELESTIAL   SPHERE  II 

resists  this  attraction,  the  actual  effect  is  not  to  permanently 
change  the  inclination  of  the  equator  to  the  orbit,  but  first  to 
cause  the  earth's  axis  to  describe  a  cone  about  an  axis  per- 
pendicular to  the  orbit,  and  second  to  cause  the  inclination  of 
the  axis  to  go  through  certain  periodic  changes  (see  Fig.  9).  The 
movement  of  the  axis  in  a  conical  surface  causes  the  line  of 
intersection  of  the  equator  and  the  plane  of  the  orbit  to  revolve 
slowly  westward,  the  pole  itself  always  moving  directly  toward 
the  vernal  equinox.  This  causes  the  equinoctial  points  to  move 
westward  in  the  sky,  and  hence  the  sun  crosses  the  equator  each 
spring  earlier  than  it  would  otherwise;  this  is  known  as  the 


-Plarferof-Eartffs-Orbit- 


FIG.  9.    PRECESSION  OF  THE  EQUINOXES 

precession  of  the  equinoxes.  In  Fig.  9  the  pole  occupies  suc- 
cessively the  positions  /,  2  and  J,  which  causes  the  point  V  to 
move  to  points  i,  2  and  j.  This  motion  is  but  50". 2  per  year, 
and  it  therefore  requires  about  25,800  years  for  the  pole  to  make 
one  complete  revolution.  The  force  causing  the  precession  is 
not  quite  constant,  and  the  motion  of  the  equinoctial  points  is 
therefore  not  perfectly  uniform  but  has  a  small  periodic  varia- 
tion. In  addition  to  this  periodic  change  in  the  rate  of  the 
precession  there  is  also  a  slight  periodic  change  in  the  obliquity, 


12 


PRACTICAL  ASTRONOMY 


called  Nutation.  The  maximum  value  of  the  nutation  is  about 
9";  the  period  is  about  19  years.  The  phenomenon  of  preces- 
sion is  clearly  illustrated  by  means  of  the  apparatus  called  the 
gyroscope.  As  a  result  of  the  precessional  movement  of  the 
axis  all  of  the  stars  gradually  change  their  positions  with  refer- 
ence to  the  plane  of  the  equator  and  the  position  of  the  equinox. 
The  stars  themselves  have  but  a  very  slight  angular  motion, 
this  apparent  change  in  position  being  due  almost  entirely  to  the 
change  in  the  positions  of  the  circles  of  reference. 

9.  Aberration  of  Light. 

Another  apparent  displacement  of  the  stars,  due  to  the  earth's 
motion,  is  that  known  as  aberration.  On  account  of  the 
rapid  motion  of  the  earth  through  space,  the  direction  in  which 
a  star  is  seen  by  an  observer  is  a  result  of  the  combined  velocities 
of  the  observer  and  of  light  from  the  star.  The  star  always 
appears  to  be  slightly  displaced  in  the  direction  in  which  the 
observer  is  actually  moving.  In  Fig.  10,  if  light  moves  from  C 
to  B  in  the  same  length  of  time  that  the  observer  moves  from 
A  to  B,  then  C  would  appear  to  be  in  the  direction  AC.  This 


FIG.  10 


FIG.  ii 


may  be  more  clearly  understood  by  using  the  familiar  illustra- 
tion of  the  falling  raindrop.  If  a  raindrop  is  falling  vertically, 
CB,  Fig.  n,  and  while  it  is  falling  a  person  moves  from  A  to  B, 
then,  considering  only  the  two  motions,  it  appears  to  the  person 
that  the  raindrop  has  moved  toward  him  in  the  direction  CA. 
If  a  tube  is  to  be  held  in  such  a  way  that  the  raindrop  shall  pass 
through  it  without  touching  the  sides,  it  must  be  held  at  the 


THE  CELESTIAL  SPHERE  13 

inclination  of  AC.  The  apparent  displacement  of  a  star  due 
to  the  observer's  motion  is  similar  to  the  change  in  the  apparent 
direction  of  the  raindrop. 

There  are  two  kinds  of  aberration,  annual  and  diurnal. 
Annual  aberration  is  that  produced  by  the  earth's  motion  in  its 
orbit  and  is  the  same  for  all  observers.  Diurnal  aberration  is 
due  to  the  earth's  daily  rotation  about  its  axis,  and  is  different 
in  different  latitudes,  because  the  speed  of  a  point  on  the  earth's 
surface  is  greatest  at  the  equator  and  diminishes  toward  the  pole. 

If  v  represents  the  velocity  of  the  earth  in  its  orbit  and  V  the 
velocity  of  light,  then  when  CB  is  at  right  angles  to  AB  the 
displacement  is  a  maximum  and 

v 

tan  OLQ  =  — , 

where  a0  is  the  angular  displacement  and  is  called  the  "constant 
of  aberration."  Its  value  is  about  20. "5.  If  CB  is  not  per- 
pendicular to  AB.  then 


or  approximately 


sin  a  =  —  sin  A 


tan  a  =  sin  a  =  —  sin  B, 


where  a  is  the  angular  displacement  and  B  is  the  angle  ABC. 

Problems 

1.  Referring  to  Fig.  2,  make  a  sketch  showing  the  path  which  Jupiter  appears 
to  describe,  in  the  plane  of  its  motion,  but  considering  the  earth  as  a  fixed  point 
on  the  diagram. 

2.  Discuss  the  approximations  made  in  the  equation  given  above  for  abbera- 
tion.     (v  is  about  i8£  miles  per  sec.;   Fis  about  186,000  miles  per  sec.) 


CHAPTER  II 

DEFINITIONS— POINTS  AND  CIRCLES  OF  REFERENCE 

10.  The  following  astronomical  terms  are  in  common  use  and 
are  necessary  in  denning  the  positions  of  celestial  objects  on  the 
sphere  by  means  of  spherical  coordinates. 

Vertical  Line. 

A  vertical  line  at  any  point  on  the  earth's  surface  is  the  direc- 
tion of  gravity  at  that  point,  and  is  shown  by  the  plumb  line 
or  indirectly  by  means  of  the  spirit  level  (OZ,  Fig.  12). 

Zenith  —  Nadir. 

If  the  vertical  at  any  point  be  prolonged  upward  it  will  pierce 
the  sphere  at  a  point  called  the  Zenith  (Z,  Fig.  12).  This  point 
is  of  great  importance  because  it  is  the  point  on  the  sphere  which 
indicates  the  position  of  the  observer  on  the  earth's  surface. 
The  point  where  the  vertical  prolonged  downward  pierces  the 
sphere  is  called  the  Nadir  (N',  Fig.  12). 

Horizon. 

The  horizon  is  the  great  circle  on  the  celestial  sphere  cut  by 
a  plane  through  the  centre  of  the  earth  perpendicular  to  the 
vertical  (NESW,  Fig.  12).  The  horizon  is  everywhere  90°  from 
the  zenith  and  the  nadir.  It  is  evident  that  a  plane  through  the 
observer  perpendicular  to  the  vertical  cuts  the  sphere  in  this 
same  great  circle.  The  visible  horizon  is  the  circle  where  the 
sea  and  sky  seem  to  meet.  Projected  onto  the  sphere  it  is  a 
small  circle  below  the  true  horizon  and  parallel  to  it.  Its  dis- 
tance below  the  true  horizon  depends  upon  the  height  of  the 
observer's  eye  above  the  surface  of  the  water*. 

Vertical  Circles. 

Vertical  Circles  are  great  circles  passing  through  the  zenith 
and  nadir.  They  all  cut  the  horizon  at  right  angles  (HZJ, 
Fig.  12). 

14 


POINTS  AND  CIRCLES  OF  REFERENCE  IS 

Almucantars. 

Parallels  of  altitude,  or  almucantars,  are  small  circles  parallel 
to  the  horizon  (DFG,  Fig.  12). 

Poles. 

If  the  earth's  axis  of  rotation  be  produced  indefinitely  it  will 
pierce  the  sphere  in  two  points  called  the  celestial  poles  (PPf 
Fig.  12). 

Equator. 

The  celestial  equator  is  a  great  circle  of  the  celestial  sphere 
cut  by  a  plane  through  the  centre  of  the  earth  perpendicular  to 


FIG.  12.    THE   CELESTIAL  SPHERE 


the  axis  of  rotation  (QWRE,  Fig.  12).  It  is  everywhere  90° 
from  the  poles.  A  parallel  plane  through  the  observer  cuts  the 
sphere  in  the  same  circle. 


1 6  PRACTICAL  ASTRONOMY 

Hour  Circles. 

Hour  Circles  are  great  circles  passing  through  the  north  and 
south  celestial  poles  (PVPf,  Fig.  12). 

Parallels  of  Declination. 

Small  circles  parallel  to  the  plane  of  the  equator  are  called 
parallels  of  decimation  (BKC,  Fig.  12). 

Meridian. 

The  meridian  is  the  great  circle  passing  through  the  zenith  and 
the  poles  (SZPL,  Fig.  12).  It  is  at  once  an  hour  circle  and  a 
vertical  circle.  It  is  evident  that  different  observers  will  in 
general  have  different  meridians.  The  meridian  cuts  the  horizon 
in  the  north  and  south  points  (N,  S,  Fig.  12).  The  intersection 
of  the  plane  of  the  meridian  with  the  horizontal  plane  through 
the  observer  is  the  meridian  line  used  in  plane  surveying. 

Prime  Vertical. 

The  prime  vertical  is  the  vertical  circle  whose  plane  is  per- 
pendicular to  the  plane  of  the  meridian  (EZW,  Fig.  12).  It 
cuts  the  horizon  in  the  east  and  west  points  (E,  W,  Fig.  12). 

Ecliptic. 

The  ecliptic  is  the  great  circle  on  the  celestial  sphere  which 
the  sun's  centre  appears  to  describe  during  one  year  (AMVL, 
Fig.  12).  Its  plane  is  the  plane  of  the  earth's  orbit;  it  is  inclined 
to  the  plane  of  the  equator  at  an  angle  of  about  23°  27',  called  the 
obliquity  of  the  ecliptic. 

Equinoxes. 

The  points  of  intersection  of  the  ecliptic  and  the  equator  are 
called  the  equinoctial  points  or  simply  the  equinoxes.  That 
intersection  at  which  the  sun  appears  to  cross  the  equator  when 
going  from  the  south  side  to  the  north  side  is  called  the  Vernal 
Equinox,  or  sometimes  the  First  Point  of  Aries  (V,  Fig.  12). 
The  sun  reaches  this  point  about  March  21.  The  other  inter- 
section is  called  the  Autumnal  Equinox  (A,  Fig.  12). 

Solstices. 

The  points  on  the  ecliptic  midway  between  the  equinoxes  are 
called  the  winter  and  summer  solstices. 


POINTS  AND  CIRCLES  OF  REFERENCE  I/ 

Colures. 

The  great  circle  through  the  poles  and  the  equinoxes  is  called 


FIG.  12.    THE  CELESTIAL  SPHERE 

the  equinoctial  colure  (PVPf,  Fig.  12).     The  great  circle  through 
the  poles  and  the  solstices  is  called  the  solstitial  colure. 

Questions 

1.  What  imaginary  circles  on  the  earth's  surface  correspond  to  hour  circles? 
To  parallels  of  declination?     To  vertical  circles? 

2.  What  are  the  widths  of  the  torrid,  temperate  and  arctic  zones  and  how  are 
they  determined? 


CHAPTER  III 

SYSTEMS   OF   COORDINATES   ON  THE   SPHERE 

ii.   Spherical  Coordinates. 

The  direction  of  a  point  in  space  may  be  denned  by  means 
of  two  spherical  coordinates,  that  is,  by  two  angular  distances 
measured  on  a  sphere  along  arcs  of  two  great  circles  which 
cut  each  other  at  right  angles.  Suppose  that  it  is  desired  to 
locate  C  (Fig.  13)  with  reference  to  the  plane  OAB  and  the  line 


FIG.  13.     SPHERICAL  COORDINATES 

OA,  0  being  the  origin  of  coordinates.  Pass  a  plane  OBC 
through  OC  perpendicular  to  OAB;  these  planes  will  intersect 
in  the  line  OB.  The  two  angles  which  fix  the  position  of  C,  or 
the  spherical  coordinates,  are  BOC  and  AOB.  These  may  be 
regarded  as  the  angles  at  the  centre  of  the  sphere  or  as  the  arcs 
BC  and  AB.  In  every  system  of  spherical  coordinates  the  two 
coordinates  are  measured,  one  on  a  great  circle  called  the  primary, 
and  the  other  on  one  of  a  system  of  great  circles  at  right  angles 
to  the  primary  called  secondaries.  There  are  an  infinite  number 
of  secondaries,  each  passing  through  the  two  poles  of  the  primary. 
The  coordinate  measured  from  the  primary  is  an  arc  of  a 

18 


SYSTEMS  OF   COORDINATES  ON  THE  SPHERE  19 

secondary  circle;  the  coordinate  measured  between  the  secondary 
circles  is  an  arc  of  the  primary. 

12.  Horizon  System. 

In  this  system  the  primary  circle  is  the  horizon  and  the  sec- 
ondaries are  vertical  circles,  or  circles  passing  through  the  zenith 
and  nadir.  The  first  coordinate  of  a  point  is  its  angular  distance 
above  the  horizon,  measured  on  a  vertical  circle;  this  is  called 
the  Altitude.  The  complement  of  the  altitude  is  called  the 
Zenith  distance.  The  second  coordinate  is  the  angular  distance 
on  the  horizon  between  the  meridian  and  the  vertical  circle 
through  the  point ;  this  is  called  the  Azimuth.  Azimuth  may  be 
reckoned  either  from  the  north  or  the  south  point  and  in  either 
direction,  like  bearings  in  surveying,  but  the  custom  is  to  reckon 
it  from  the  south  point  right-handed  from  o°  to  360°  except  for 
stars  near  the  pole,  in  which  case  it  is  more  convenient  to  reckon 


—  W—          Azimuth 

FIG.  14.    THE  HORIZON  SYSTEM 

from  the  north,  and  either  to  the  east  or  to  the  west.  In  Fig.  14 
the  altitude  of  the  star  A  is  BA ;  its  azimuth  is  SB. 

13.  The  Equator  Systems. 

The  circles  of  reference  in  this  system  are  the  equator  and 
great  circles  through  the  poles,  or  hour  circles.  The  first  coor- 
dinate of  a  point  is  its  angular  distance  north  or  south  of  the 


20 


PRACTICAL  ASTRONOMY 


equator,  measured  on  an  hour  circle;  it  is  called  the  Decimation. 
Declinations  are  considered  positive  when  north  of  the  equator, 
negative  when  south.  The  complement  of  the  declination  is 
called  the  Polar  Distance.  The  second  coordinate  of  the  point 
is  the  arc  of  the  equator  between  the  vernal  equinox  and  the  foot 
of  the  hour  circle  through  the  point;  it  is  called  Right  Ascension. 
Right  ascension  is  measured  from  the  equinox  eastward  to  the 
hour  circle  through  the  point  in  question ;  it  may  be  measured  in 
degrees,  minutes,  and  seconds  of  arc,  or  in  hours,  minutes,  and 


FIG.  15.    THE  EQUATOR  SYSTEM 

seconds  of  time.    In  Fig.  15  the  declination  of  the  star  S  is  AS; 
the  right  ascension  is  VA. 

Instead  of  locating  a  point  by  means  of  declination  and  right 
ascension  it  is  sometimes  more  convenient  to  use  declination 
and  Hour  Angle.  The  hour  angle  of  a  point  is  the  arc  of  the 


SYSTEMS  OF  COORDINATES  ON  THE  SPHERE 


21 


equator  between  the  observer's  meridian  and  the  hour  circle 
through  the  point.  It  is  measured  from  the  meridian  westward 
(clockwise)  from  oh  to  24^  or  from  o°  to  360°.  In  Fig.  16  the 
declination  of  the  star  S  is  AS  (negative);  the  hour  angle  is 


M 


FIG.  16.    HOUR  ANGLE  AND  DECLINATION 

M A .     It  is  evident  that  in  general  the  hour  angles  of  all  points 
on  the  celestial  sphere  are  always  increasing. 

These  three  systems  are  shown  in  the  following  table. 


Name. 

Primary. 

Secondaries. 

Origin  of 
Coordinates. 

ist  coord. 

and  coord. 

Horizon  System 

Horizon 
Equator 

Vert.  Circles 
Hour  Circles 

South  point. 
Vernal  Equi- 

Altitude 
Declin. 

Azimuth 
Rt.  Ascen. 

nox. 

Equator  Systems  • 

<( 

tt        « 

Intersection 

« 

Hour  Angle 

of  Meridian 

and  Equator. 

22 


PRACTICAL  ASTRONOMY 


14.  There  is  another  system  which  is  employed  in  some 
branches  of  astronomy  but  will  not  be  used  in  this  book.     The 
coordinates  are  called  celestial  latitude  and  celestial  longitude  j 
the  primary  circle  is  the  ecliptic.     Celestial  latitude  is  measured 
from  the  ecliptic  just  as  declination  is  measured  from  the  equator. 
Celestial  longitude  is  measured  eastward  along  the  ecliptic  from 
the  equinox,  just  as  right  ascension  is  measured  eastward  along 
the  equator.    The  student  should  be  careful  not  to  confuse  celes- 
tial latitude  and  longitude  with  terrestrial  latitude  and  longitude. 
The  latter  are  the  ones  used  in  the  problems  discussed  in  this  book. 

15.  Coordinates  of  the  Observer. 

The  observer's  position  is  located  by  means  of  his  latitude  and 
longitude.  The  latitude,  which  on  the  earth's  surface  is  the 
angular  distance  of  the  observer  north  or  south  of  the  equator, 
may  be  defined  astronomically  as  the  declination  of  the  ob- 
server's zenith.  In  Fig.  17,  the  terrestrial  latitude  is  the  arc  E0> 


p' 


FIG.  17.    THE  OBSERVER'S  LATITUDE 

EQ  being  the  equator  and  O  the  observer.  The  point  Z  is  the 
observer's  zenith,  so  that  the  latitude  on  the  sphere  is  the  arc 
E'Z,  which  evidently  will  contain  the  same  number  of  degrees 
as  EO.  The  complement  of  the  latitude  is  called  the  Co-latitude. 


SYSTEMS   OF   COORDINATES   ON  THE   SPHERE  23 

The  terrestrial  longitude  of  the  observer  is  the  arc  of  the  equator 
between  the  primary  meridian  (usually  that  of  Greenwich)  and 
the  meridian  of  the  observer.  On  the  celestial  sphere  the  longi- 
tude would  be  the  arc  of  the  celestial  equator  contained  between 
two  hour  circles  whose  planes  are  the  planes  of  the  two  terrestrial 
meridians. 

1 6.  Relation  between  the  Two  Systems  of  Coordinates. 

In  studying  the  relation  between  different  points  and  circles 
on  the  sphere  it  may  be  convenient  to  imagine  that  the  celestial 
sphere  consists  of  two  spherical  shells,  one  within  the  other. 


FIG.  1 8.    THE  SPHERE  SEEN  FROM  THE  OUTSIDE 

The  outer  one  carries  upon  its  surface  the  ecliptic,  equinoxes, 
poles,  equator,  hour  circles  and  all  of  the  stars,  the  sun,  the  moon 
and  the  planets.  On  the  inner  sphere  are  the  zenith,  horizon, 
vertical  circles,  poles,  equator,  hour  circles,  and  the  meridian. 
The  earth's  daily  rotation  causes  the  inner  sphere  to  revolve, 


24  PRACTICAL  ASTRONOMY 

while  the  outer  sphere  is  motionless,  or,  regarding  only  the 
apparent  motion,  the  outer  sphere  revolves  once  per  day  on  its 
axis,  while  the  inner  sphere  appears  to  be  motionless.  It  is 
evident  that  the  coordinates  of  a  fixed  star  in  the  first  equatorial 
system  (Declination  and  Right  Ascension)  are  practically  always 
the  same,  whereas  the  coordinates  in  the  horizon  system  are 
continually  changing.  It  will  also  be  seen  that  in  the  first 
equatorial  system  the  coordinates  are  independent  of  the  ob- 
server's position,  but  in  the  horizon  system  they  are  entirely 
dependent  upon  his  position.  In  the  second  equatorial  system 
one  coordinate  is  independent  of  the  observer,  while  the  other 
(hour  angle)  is  not.  In  making  up  catalogues  of  the  positions 
of  the  stars  it  is  necessary  to  use  right  ascensions  and  declina- 
tions in  defining  these  positions.  When  making  observations 


FIG.  19.    PORTION  OF  THE  SPHERE  SEEN  FROM  THE  EARTH  (LOOKING  SOUTH) 


with  instruments  it  is  usually  simpler  to  measure  coordinates 
in  the  horizon  system.  Therefore  it  is  necessary  to  be  able  to 
compute  the  coordinates  of  one  system  from  those  of  another. 
The  mathematical  relations  between  the  spherical  coordinates 
are  discussed  in  Chapter  IV. 


SYSTEMS  OF   COORDINATES  ON  THE  SPHERE  25 

Figs.  1 8,  19,  and  20  show  three  different  views  of  the  celestial 
sphere  with  which  the  student  should  be  familiar.  Fig.  18  is 
the  sphere  as  seen  from  the  outside  and  is  the  view  best  adapted 
to  showing  problems  in  spherical  trigonometry.  The  star  S  has 
the  altitude  RS,  azimuth  S'R,  hour  angle  Mm,  right  ascension 
Vm}  and  declination  mS\  the  meridian  is  ZMS' '.  Fig.  19  shows 
a  portion  of  the  sphere  as  seen  by  an  observer  looking  southward; 
the  points  are  indicated  by  the  same  letters  as  in  Fig.  18.  Fig.  20 


FIG.  20.    THE  SPHERE  PROJECTED  ONTO  THE  PLANE  or  THE  EQUATOR 


shows  the  same  ^points  projected  on  the  plane  of  the  equator. 
In  this  view  of  the  sphere  the  angles  at  the  pole  (i.e.,  the 
angles  between  hour  circles)  are  shown  their  true  size,  and 
it  is  therefore  a  convenient  diagram  to  use  when  dealing  with 
right  ascension  and  hour  angles. 


26  PRACTICAL  ASTRONOMY 


Questions  and  Problems 

1.  What  coordinates  on  the  sphere  correspond  to  latitude  and  longitude  on  the 
earth's  surface? 

2.  Make  a  sketch  of  the  sphere  and  plot  the  position  of  a  star  having  an  altitude 
of  20°  and  an  azimuth  of  250°.     Locate  a  star  whose  hour  angle  is  i6h  and  whose 
declination  is   — 10°.     Locate  a  star  whose  right  ascension  is  9^  and  whose  declina- 
tion is  N.  30°. 

3.  If  a  star  is  on  the  equator  and  also  on  the  horizon,  what  is  its  azimuth?     Its 
altitude?     Its  hour  angle?     Its  declination? 

4.  What  changes  take  place  in  the  azimuth  and  altitude  of  a  star  during 
twenty-four  hours  ? 

5.  What  changes  take  place  in  the  right  ascension  and  declination  of  the  ob- 
server's zenith  during  a  day  ? 

6.  A  person  in  latitude  40°  N.  observes  a  star,  in  the  west,  whose  declination  is 
5°  N. .  In  what  order  will  the  star  pass  the  following  three  circles;  (a)  the  6h  circle, 
(b)  the  horizon,  (c)  the  prime  vertical  ? 


CHAPTER  IV 


RELATION   BETWEEN   COORDINATES 

17.   Relation  between  Altitude  of  Pole  and  Latitude  of  Ob- 
server. 

In  Fig.  21,  SZN  represents  the  observer's  meridian;  let  P  be 
the  celestial  pole,  Z  the  zenith,  E  the  point  of  intersection  of  the 
meridian  and  the  equator,  and 
N  and  S  the  north  and  south 
points  of  the  horizon.  By  the 
definitions,  OZ  (vertical)  is 
perpendicular  to  SN  (horizon) 
and  OP  (axis)  is  perpendicular 
to  EO  (equator).  Therefore 
the  arc  PN  =  arc  EZ.  By  the 


E 


FIG.  22 

definitions,  EZ  is  the  declination  of  the  zenith,  or  the  latitude, 
and  PN  is  the  altitude  of  the  celestial  pole.  Hence  the  altitude 
of  the  pole  is  always  equal  to  the  latitude  of  the  observer.  The  same 
relation  maybe  seen  from  Fig.  22,  in  which  P  is  the  north  pole 
of  the  earth,  OH  is  the  plane  of  the  horizon,  the  observer  being 
at  Oj  EQ  is  the  earth's  equator,  and  OP'  is  a  line  parallel  to  CP 
and  consequently  points  to  the  celestial  pole.  It  may  readily 
be  shown  that  ECO,  the  observer's  latitude,  equals  HOP',  the 
altitude  of  the  celestial  pole.  A  person  at  the  equator  would 

27 


28 


PRACTICAL  ASTRONOMY 


see  the  north  celestial  pole  in  the  north  point  of  his  horizon  and 
the  south  celestial  pole  in  the  south  point  of  his  horizon.  If  he 
travelled  northward  the  north  pole  would  appear  to  rise,  its 
altitude  being  always  equal  to  his  latitude,  while  the  south  pole 
would  immediately  go  below  his  horizon.  When  the  traveller 
reached  the  north  pole  of  the  earth  the  north  celestial  pole 
would  be  vertically  over  his  head. 

To  a  person  at  the  equator  all  stars  would  appear  to  move 
vertically  at  the  times  of  rising  and  setting,  and  all  stars  would 
be  above  the  horizon  i2h  and  below  i2h  during  one  revolution 


(S.Pole) 


N  (N.Pole) 


FIG.  23.    THE  RIGHT  SPHERE 

of  the  sphere.  All  stars  in  both  hemispheres  would  be  above 
the  horizon  at  some  time  every  day.  This  is  called  the  "  right 
sphere"  (Fig.  23). 

If  a  person  were  at  the  earth's  pole  the  celestial  equator  would 
coincide  with  his  horizon,  and  all  stars  in  the  northern  hemi- 
sphere would  appear  to  travel  around  in  circles  parallel  to  the 
horizon;  they  would  be  visible  for  24^  a  day,  and  their  altitudes 
would  not  change.  The  stars  in  the  southern  hemisphere  would 
never  be  visible.  The  word  north  would  cease  to  have  its  usual 


RELATION  BETWEEN  COORDINATES  29 

meaning,  and  south  might  mean  any  horizontal  direction.  The 
longitude  of  a  point  on  the  earth  and  its  azimuth  from  the 
Greenwich  meridian  would  then  be  the  same.  This  is  called 
the  "parallel  sphere"  (Fig.  24). 

At  all  points  between  these  two  extreme  latitudes  the  equator 
cuts  the  horizon  obliquely.    A  star  on  the  equator  will  be  above 


FIG.  24.    THE  PARALLEL  SPHERE 

the  horizon  half  the  time  and  below  half  the  time.  A  star  north 
of  the  equator  will  (to  a  person  in  the  northern  hemisphere)  be 
above  the  horizon  more  than  half  of  the  day;  a  star  south  of  the 
equator  will  be  above  the  horizon  less  than  half  of  the  day.  If 
the  north  polar  distance  of  a  star  is  less  than  the  observer's  north 
latitude,  the  whole  of  the  star's  diurnal  circle  is  above  the  hori- 
zon, and  the  star  will  therefore  remain  above  the  horizon  all 
of  the  time.  It  is  called  in  this  case  a  circumpolar  star  (Fig. 
25).  The  south  circumpolar  stars  are  those  whose  south  polar 
distances  are  less  than  the  latitude;  they  are  never  visible  to  an 
observer  in  the  northern  hemisphere.  If  the  observer  travels 


PRACTICAL  ASTRONOMY 


north  until  he  is  beyond  the  arctic  circle,  latitude  66°  33'  north, 
then  the  sun  becomes  a  circumpolar  at  the  time  of  the  summer 
solstice.  At  noon  the  sun  would  be  at  its  maximum  altitude; 
at  midnight  it  would  be  at  its  minimum  altitude  but  would  still 
be  above  the  horizon.  This  is  called  the  "  midnight  sun." 

z 


Circumpola 
(Never  Rise) 


P'' 


FIG.  25.     CIRCUMPOLAR  STARS 

18.  Relation  between  Latitude  of  Observer  and  the  Declina- 
tion and  Altitude  of  a  Star  on  the  Meridian. 

The  relation  between  the  latitude,  altitude,  and  declination 
at  the  instant  when  a  star  is  crossing  the  observer's  meridian  may 
be  seen  from  Fig.  26.  Let  A  be  a  star  on  the  meridian,  south  of 
the  zenith  and  north  of  the  equator;  then 

EZ  =  L,  the  latitude, 

EA  =  D,  the  declination, 

SA  =  k,  the  altitude, 

ZA  =  z,  the  zenith  distance. 


From  the  figure 

or 

and 

also 


ZA  =  EZ  -  EA 
z  =  L-  D 
h  =  90°  -  (L  -  Z)); 
L  =  90°  -  (h  -  D). 


[i] 

[2] 


RELATION  BETWEEN  COORDINATES  31 

If  A  is  south  of  the  equator  the  declination  is  considered 
negative,  so  the  same  equations  will  hold  true  for  this  case. 


FIG.  26.     STAR  ON  THE  MERIDIAN 

If  the  star  is  north  of  the  zenith,  as  at  B,  it  will  be  more  con- 
venient to  use  the  polar  distance,  p  =  90°  —  D. 

In  this  case  NP  =  NB  -  PB 

L  =  h-p.  [3] 


or 


If  B  is  below  the  pole  the  equation  is 

L  =  h  +  p. 


[4] 


19.  The  Astronomical  Triangle. 

By  joining  the  pole,  zenith,  and  any  star  S  on  the  sphere 
by  arcs  of  great  circles  we  obtain  a  triangle  from  which  the  rela- 
tion existing  among  the  spherical  coordinates  may  be  obtained. 
This  triangle  is  so  frequently  employed  in  astronomy  and  navi- 
gation that  is  it  called  the  "astronomical  triangle"  or  the  "PZS 
triangle."  In  Fig.  27  the  arc  PZ  is  the  complement  of  the 
latitude,  or  co-latitude ;  arc  ZS  is  the  zenith  distance  or  comple- 
ment of  the  altitude;  arc  PS  is  the  polar  distance  or  complement 
of  the  declination;  the  angle  P  is  the  hour  angle  of  the  star  if 
5  is  west  of  the  meridian,  or  360°  minus  the  hour  angle  if 
S  is  east  of  the  meridian;  and  Z  is  the  azimuth  of  5,  or  360° 
minus  the  azimuth,  according  as  S  is  west  or  east  of  the  meridian. 
The  angle  at  5  is  called  the  parallactic  angle;  it  is  little  used  in 
practical  astronomy.  If  any  three  parts  of  this  triangle  are 


32  PRACTICAL  ASTRONOMY 

known  the  other  three  may  be  calculated.      The  fundamental 
formulae  of  spherical  trigonometry  are 

cos  a  =  cos  b  cos  c  +  sin  b  sin  c  cos  A,  [5] 

sin  a  cos  B  =  cos  &  sin  c  —  sin  6  cos  c  cos  ^4,  [6] 

sin  a  sin  £  =  sin  b  sin  ^4.  [7] 

If  we  put  A  =  P,  B  =  S,  C  =  Z,  a  =  90°  -  h,  b  =  90°  -  L, 
c  =  90°  —  D,  then  these  become 

sin  h  =  sin  L  sin  D  +  cos  L  cos  D  cos  P,  [8] 

cos  h  cos  5  =  sin  L  cos  Z)  —  cos  L  sin  Z)  cos  P,  [9] 

cos  h  sin  5  =  cos  L  sin  P.  [10] 

If  A  =  P,B  =  Z,C  =  S,a  =  9o0-h,b  =  9o°-D,c  =  9o°-L, 
then 

cos  h  cos  Z  =  sin  Z)  cos  L  —  cos  Z?  sin  L  cos  P,  [u] 

cos  h  sin  Z  =  cos  Z)  sin  P.  [12] 

If  A  =  Z,  B  =  S,  C  =  P,  a  =  90°  -  D,  b  =  90°  -  L,  c  =  90°  -  h, 
then 

sin  D  =  sin  L  sin  h  +  cos  L  cos  h  cos  Z,  [13] 

cos  D  cos  5  =  sin  L  cos  ^  —  cos  L  sin  /z  cos  Z,  [14] 

cos  Z)  sin  S  =  cos  L  sin  Z.  [15] 

Other  forms  may  be  derived  by  assigning  different  values  to 
the  parts  of  the  triangle  ABC.  The  formulse  given  in  the 
following  chapters  may  in  nearly  all  cases  be  derived  from 
equations  [5]  to  [15]. 

The  most  common  cases  arising  in  the  practice  of  surveying 
are:  — 

1.  Given  the  declination,  latitude,  and  altitude,  to  find  the 
azimuth  and  the  hour  angle. 

2.  Given  the  declination,  latitude,  and  hour  angle,  to  find  the 
azimuth  and  the  altitude. 


RELATION  BETWEEN  COORDINATES 


33 


In  the  following  formulae 


let 


and  also  let 


P  =  the  hour  angle, 
Z  =  the  azimuth,* 

h  =  the  altitude, 

z  =  the  zenith  distance, 
D  =  the  declination, 

p  =  the  polar  distance, 
L  =  the  latitude, 

5  =  ±(h  +  L  +  p). 


FIG.  27.    THE  ASTRONOMICAL  TRIANGLE 

For  computing  P  any  of  the  following  formulae  may  be  used. 
a  |  [z •+  (L  —  D)}  sin^  [z  —  (L  — 


cos  L  cos  D 


*  In  the  formulae  which  follow  Z  is  reckoned  from  the  north  (interior  angle) 
unless  otherwise  designated. 


34  PRACTICAL  ASTRONOMY 


•        i      •/•>  »   /  I  ^**»  «*   oiii  \o     —   »»'/  \  r        i 

sinJP=y   -      — — ^ — —-M.  [17] 

r     V         ^AAO    /     p/~)c   /"l         ' 


COS 


IP-  .    /cos  (s  -  p)  sin  (s  -  L)\  ,    , 

"VI          cosPcosi          /• 


-        cos  s  — 


cos  L  cos 


„      sin  ^  —  sin  Z,  sin  Z>  r     , 

cosP  =  -         —  —  -.  21] 

cos  Z,  cos  Z) 

For  computing  the  angle  Z  (measured  from  the  north  point)  we 
have 


-  D)  cosj  (,  +  L  +  D) 


cos  L  sin 


\       { 
/ 


sin|Z=V(-n^-^sm^-^).  [23] 

cos  L  —  7"  ' 


cosjz  =  v  -    -;v"  /'].  N] 

'  •      —  r  cos  h 


[25] 


cos  s  cos  (s  —  p 

.      •  [26] 


cos     cos  « 

„      sin  Z)  —  sin  L  sin  &  r    , 

cos  Z  =  --  -  --  -  -  .  [27] 

cos  L  cos  h 

While  any  of  these  formulae  may  be  used  to  determine  the  angle 
sought,  the  choice  of  formula  should  depend  somewhat  upon 

*  In  this  case  Z  is  reckoned  from  the  south. 


RELATION  BETWEEN   COORDINATES  35 

the  precision  with  which  the  angle  is  denned  by  the  function. 
If  the  angle  is  quite  small  it  is  more  accurately  found  through  its 
sine  than  through  its  cosine;  for  an  angle  near  90°  the  reverse 
is  the  case.  On  account  of  the  rapid  variation  of  the  tangent 
an  angle  is  always  more  precisely  determined  by  this  function 
than  by  either  the  sine  or  the  cosine.  The  last  two  formulae  in 
each  set  require  the  use  of  both  natural  and  logarithmic  func- 
tions, but  are  sometimes  convenient. 

The  altitude  may  be  found  from  the  formulae 

sin  h  =  cos  (L  —  D)  —  2  cos  L  cos  D  sin2  J  P         [28] 
or  sin  h  =  cos  (L  —  D)  —  cos  L  cos  D  vers  P,  [29] 

which  may  be  derived  from  Equa.  [8]. 

If  the  declination,  hour  angle,  and  altitude  are  given,  the 
azimuth  is  found  from 

.     „        .     n  cos  D 
sin  Z  =  sm  P  - 

cos  h 

=  sin  P  cos  D  sec  h.  [30] 

For  computing  the  azimuth  of  a  star  near  the  pole  when  the 
hour  angle  is  known  the  following  formula  is  frequently  used: 


sn 


/ 
'      / 


This  equation  may  be  derived  by  dividing  [12]  by  [n]  and  then 
simplifying  the  result  by  dividing  by  cos  D. 

Given  the  latitude  and  declination,  find  the  hour  angle  and 
azimuth  of  a  star  on  the  horizon.  Putting  h  =  o  in  Equa.  [8] 
and  [13]  the  results  are 

cos  P  =  —  tan  D  tan  L  [32] 

sin  D  P    , 

[33] 


36  PRACTICAL  ASTRONOMY 

A  special  case  of  the  PZS  triangle  occurs  when  a  star  which 
culminates  north  of  the  zenith  is  at  its  farthest  east  or  west  posi- 
tion, known  as  its  greatest  elongation.  At  this  time  the  star's 
azimuth  is  a  maximum  and  its  diurnal  circle  is  tangent  to  the 


HORIZON  A  N 

FIG.  28.    STAR  AT  GREATEST  ELONGATON  (EAST) 

vertical  circle  through  the  star  (Fig.  28)  ;  the  triangle  is  conse- 
quently right-angled  at  S. 
The  formulae  for  this  case  are 

tan  L 


and  sin  Z  =  sin  p  sec  L.  [35] 

20.  Relation  between  Right  Ascension  and  Hour  Angle, 

In  order  to  understand  the  relation  between  the  right  ascen- 
sion and  the  hour  angle  of  a  point,  we  may  think  of  the  equator 
on  the  outer  sphere  as  graduated  into  hours,  minutes,  and  seconds 
of  right  ascension,  zero  being  at  the  equinox  and  the  numbers 
increasing  toward  the  east.  The  equator  on  the  inner  sphere  is 
graduated  for  hour  angles,  the  zero  being  at  the  observer's 
meridian  and  the  numbers  increasing  toward  the  west.  (See 
Fig,  29.)  As  the  outer  sphere  turns,  the  hour  marks  on  the  right 
ascension  scale  will  pass  the  meridian  in  the  order  of  the  numbers. 
The  number  opposite  the  meridian  at  any  instant  shows  how  far 


RELATION   BETWEEN   COORDINATES 


37 


FIG.  29.    RIGHT  ASCENSION  AND  HOUR  ANGLE 


FIG.  30 


38  PRACTICAL  ASTRONOMY 

the  sphere  has  turned  since  the  equinox  was  on  the  meridian. 
If  we  read  the  hour  angle  scale  opposite  the  equinox,  we  obtain 
exactly  the  same  number  of  hours.  This  number  of  hours  (or 
angle)  may  be  considered  as  either  the  right  ascension  of  the 
meridian  or  the  hour  angle  of  the  equinox.  In  Fig.  30  the  star 
S  has  an  hour  angle  equal  to  AB  and  a  right  ascension  CB.  The 
sum  of  these  two  angles  is  AC,  or  the  hour  angle  of  the  equinox. 
The  same  relation  will  be  found  to  hold  true  for  all  positions  of 
5.  The  general  relation  existing  between  these  coordinates  is, 
then, 

Hour  angle  of  Equinox  =  Hour  angle  of  Star  +  Right  A  seen- 
sion  of  Star.  [36] 

Questions  and  Problems 

1.  What  is  the  greatest  declination  a  star  may  have  and  pass  the  meridian 
to  the  south  of  the  zenith? 

2.  What  angle  does  the  plane  of  the  equator  make  with  the  horizon? 

3.  In  what  latitudes  can  the  sun  be  overhead? 

4.  What  is  the  altitude  of  the  sun   at  noon   hi  Boston    (42°  21'   N.)    on 
December  22? 

5.  What  are  the  greatest  and  least  angles  made  by  the  ecliptic  with  the  hori- 
zon at  Boston? 

6.  In  what  latitudes  is  Vega  (Decl.  =  38°  42'  N.)  a  circumpolar  star? 

7.  Make  a  sketch  of  the  celestial  sphere  like  Fig.  12  corresponding  to  a  lati- 
tude of  20°  south  and  the  instant  when  the  vernal  equinox  is  on  the  eastern 
horizon. 

8.  Derive  formula  [35]. 

9.  Compute  the  hour  angle  of  Vega  when  it  is  rising  in  latitude  40°  North. 
10.   Compute  the  time  of  sunrise  on  June  22,  in  latitude  40°  N. 


CHAPTER  V 
MEASUREMENT  OF  TIME 

21.  The  Earth's  Rotation. 

The  measurement  of  intervals  of  time  is  made  to  depend  upon 
the  period  of  the  earth's  rotation  on  its  axis.  Although  it  is 
probable  that  this  period  is  not  absolutely  invariable,  yet  the 
variations  are  too  small  to  be  measured,  and  the  rotation  is 
assumed  to  be  uniform.  The  most  natural  unit  of  time  for 
ordinary  purposes  is  the  solar  day,  or  the  time  of  one  rotation 
of  the  earth  with  respect  to  the  sun's  direction.  On  account  of 
the  earth's  annual  motion  around  the  sun  the  direction  of  the 
reference  line  is  continually  changing,  and  the  length  of  the 
solar  day  is  not  the  true  time  of  one  rotation  of  the  earth  on  its 
axis.  For  this  reason  it  is  necessary  in  astronomical  work  to 
make  use  of  another  kind  of  time,  based  upon  the  actual  period  of 
rotation,  called  sidereal  time  (star  time). 

22.  Transit  or  Culmination. 

Every  point  on  the  celestial  sphere  crosses  the  meridian  of  an 
observer  twice  during  one  revolution  of  the  sphere.  The  instant 
when  any  point  on  the  celestial  sphere  is  on  the  meridian  of  an 
observer  is  called  the  transit,  or  culmination,  of  that  point  over 
that  meridian.  When  it  is  on  that  half  of  the  meridian  contain- 
ing the  zenith,  it  is  called  the  upper  transit;  when  it  is  on  the 
other  half  it  is  called  the  lower  transit.  Except  in  the  case  of 
stars  near  the  elevated  pole  the  upper  transit  is  the  only  one 
visible  to  the  observer;  hence  when  the  transit  of  a  star  is  men- 
tioned the  upper  transit  will  be  understood  unless  the  contrary 
is  stated. 

23.  Sidereal  Day. 

The  sidereal  day  is  the  interval  of  time  between  two  successive 
upper  transits  of  the  vernal  equinox  over  the  same  meridian. 

39 


40  PRACTICAL   ASTRONOMY 

If  the  equinox  were  absolutely  fixed  in  position,  the  sidereal  day 
as  thus  defined  would  be  the  true  period  of  the  earth's  rotation; 
but  since  the  equinox  has  a  slow  westward  motion  caused  by  the 
precessional  movement  of  the  axis  (see  Art.  8),  the  actual 
interval  between  two  transits  of  the  equinox  differs  about 
os.oi  from  the  true  time  of  one  rotation.  The  sidereal  day  actu- 
ally used  in  practice,  however,  is  the  one  defined  above  and  not 
the  true  rotation  period.  Sidereal  days  are  not  used  for  reckon- 
ing long  periods  of  time,  dates  always  being  in  solar  days,  so  this 
error  never  becomes  appreciable.  The  sidereal  day  is  divided 
into  24  hours  and  each  hour  is  subdivided  into  minutes  and 
seconds.  When  the  equinox  is  at  upper  transit  it  is  oh,  or  the 
beginning  of  the  sidereal  day  (sidereal  "  noon  "). 

24.  Sidereal  Time. 

The  sidereal  time  at  a  given  meridian  at  any  instant  is  the 
hour  angle  of  the  vernal  equinox.  It  is  therefore  a  measure  of 
the  angle  through  which  the  earth  has  turned  since  the  equinox 
was  on  the  meridian,  and  shows  the  position  of  the  sphere  at 
the  given  instant  with  respect  to  the  observer's  meridian. 

25.  Solar  Day. 

A  solar  day  is  the  interval  of  time  between  two  successive  upper 
transits  of  the  sun's  centre  over  the  same  meridian.  It  is  divided 
into  24  hours,  each  hour  being  divided  into  minutes  and  seconds. 
When  the  sun  is  on  the  upper  side  of  the  meridian  (upper 
transit)  it  is  noon,  or  oh  solar  time.  When  it  is  on  the  lower  side 
of  the  meridian  it  is  midnight. 

26.  Solar  Time. 

The  solar  time  at  a  given  meridian  at  any  instant  is  the  hour 
angle  of  the  sun's  centre  at  that  instant.  This  hour  angle  is  a 
measure  of  the  angle  through  which  the  earth  has  turned  with 
respect  to  the  sun's  direction,  and  consequently  is  a  measure  of 
the  time  elapsed  since  the  sun  was  on  the  meridian. 

Since  the  earth  revolves  around  the  sun  in  an  elliptical  orbit 
in  accordance  with  the  law  of  gravitation,  the  apparent  angular 
motion  of  the  sun  is  not  uniform,  and  the  days  are  therefore  of 


MEASUREMENT  OF  TIME  41 

unequal  length  at  different  seasons.  In  former  times,  when  sun 
dials  were  considered  sufficiently  accurate  for  measuring  time, 
this  lack  of  uniformity  was  not  important.  Under  modern 
conditions,  which  demand  accurate  measurement  of  time  by  the 
use  of  clocks,  an  invariable  unit  of  time  is  essential.  As  a  con- 
sequence, the  time  adopted  for  common  use  is  that  kept  by  a 
fictitious  sun,  or  mean  sun,  which  is  conceived  to  move  at  a 
uniform  rate  along  the  equator,*  its  speed  being  such  that  it 
makes  one  apparent  revolution  around  the  earth  in  the  same  time 
as  the  true  sun  (i.e.,  one  year).  The  fictitious  sun  is  so  placed 
that  on  the  whole  it  precedes  the  true  sun  as  much  as  it  follows 
it.  The  time  indicated  by  the  position  of  the  mean  sun  is  called 
mean  solar  time,  or  simply  mean  time.  The  time  indicated  by 
the  position  of  the  real  sun  is  called  apparent  solar  time  and  is 
the  time  shown  by  a  sun  dial. 

27.   Equation  of  Time. 

Since  observations  made  on  the  sun  for  the  purpose  of  deter- 
mining the  time  can  give  apparent  time  only,  it  is  necessary  to  be 
able  to  find  at  any  instant  the  exact  relation  between  apparent 
and  mean  time.  The  difference  between  the  two,  which  varies 
from  —  i4m  to  +  i6m  (nearly),  is  called  the  equation  of  time. 
This  quantity  may  be  found  in  the  Nautical  Almanac  for  each 
day  of  the  year. 

This  difference  between  the  two  kinds  of  time  is  due  to  several 
causes,  the  chief  of  which  are  (i)  the  inequality  of  the  earth's 
angular  motion  in  the  orbit,  and  (2)  the  fact  that  the  true  sun 
is  on  the  ecliptic  while  the  mean  sun  is  on  the  equator.  In  the 
winter,  when  the  earth  is  nearest  the  sun,  the  rate  of  angular 
motion  about  the  sun  must  be  greater  than  in  summer  in  order 
that  the  radius  vector  shall  describe  equal  areas  in  equal  inter- 
vals of  time.  (See  Fig.  6  and  Art.  6.)  The  sun  will  then  appear 

*  This  statement  is  true  in  a  general  way,  but  the  motion  is  not  strictly  uniform 
because  the  motion  of  the  equinox  itself  is  variable.  The  angle  from  the  equinox 
to  the  "  mean  sun  "  at  any  instant  is  the  sun's  "  mean  longitude  "  (along  the 
ecliptic)  plus  periodic  terms. 


42  PRACTICAL  ASTRONOMY 

to  move  eastward  in  the  sky  at  a  faster  rate  than  in  summer, 
and  its  daily  revolution  about  the  earth  will  be  slower.  This 
delays  the  instant  of  apparent  noon,  making  the  apparent  solar 
days  longer  than  their  average,  and  therefore  a  sun  dial  will 
"  lose  time."  About  April  i  the  sun  is  moving  at  its  average 
speed  and  the  sun  dial  ceases  to  lose  time;  from  this  date  until 
about  July  i  the  sun  dial  gains  on  mean  time,  making  up  what 
it  lost  between  Jan.  i  and  April  i.  During  the  other  half  of  the 
year  the  process  is  reversed;  the  sun  dial  gains  from  July  i  to 
Oct.  i  and  loses  from  Oct.  i  to  Jan.  i.  The  maximum  difference 
in  time  due  to  this  cause  is  about  8  minutes,  either  +  or  — . 

The  second  cause  of  the  equation  of  time  is  illustrated  by 
Fig.  31.     Assume  that  point  S'  (sometimes  called  the  "  first 

p 


mean  sun")  moves  uniformly  along  the  ecliptic  at  the  average 
rate  of  the  true  sun;  the  time  as  indicated  by  this  point  will 
evidently  not  be  affected  by  the  eccentricity  of  the  orbit.  If 
the  mean  sun  S  (also  called  "  the  second  mean  sun")  starts  at 
V,  the  equinox,  at  the  same  instant  that  5"  starts,  then  the  arcs 
VS  and  VS'  are  equal,  since  both  points  are  moving  with  the 
same  speed.  By  drawing  hour  circles  through  these  two  points 
it  will  be  seen  that  these  hour  circles  do  not  coincide  except 
when  the  points  are  at  the  equinoxes  or  at  the  solstices.  Since 
the  points  are  not  on  the  same  hour  circle  they  will  not  cross  the 
meridian  at  the  same  time,  the  difference  in  time  being  repre- 


MEASUREMENT  OF  TIME 


43 


sented  by  the  arc  aS.  The  maximum  length  of  aS  is  about 
10  minutes  of  time,  which  may  be  either  +  or  — .  The  com- 
bined effect  of  these  two  causes,  or  the  equation  of  time,  is 
shown  in  the  following  table. 

TABLE  A.     EQUATION  OF  TIME   FOR   1910. 


ISt. 

ioth. 

20th. 

30th. 

January  

—     3™  26* 

-    7W  27s 

—   IIm   02* 

—  I3W    22s 

February 

—  1  3       4.1 

—  14     24 

—  13     59 

March 

AO        tL 
—  12       ^8 

—  10     36 

-    7     48 

—    4     45 

April  

—    4      08 

—    I     31 

+    o     58 

+    2     47 

]May 

+      2        <^ 

•4—  3     42 

+    3     42 

+    2     48 

June 

+      2        11 

-f-    o     57 

-    i     08 

—    3     15 

July 

—      ^        27 

—    «;     01 

-    6     06 

-    6     16 

August          .        .... 

-    6     ii 

—    5     19 

-    3      26 

.  —    o     46 

September  
October     

—    o     09 
+  10     05 

+      2       48 

+  12     45 

+    6     20 
+  15     oi 

+    9     46 
+  16     13 

November  
December  

-f  16     18 
+  ii     06 

+  16     02 
+    7      23 

+  14     26 

+      2       36 

+  ii      28 

—      2        21 

January  February    Mnrch       April 


July       August  Septembe^  October   fiovember{Decemher 


•-V' 


£ 


FIG.  3ia. 

28.  Conversion  of  Mean  Time  into  Apparent  Time  and  vice 
versa. 

Mean  time  may  be  converted  into  apparent  time  by  adding 
algebraically  the  equation  of  time  for  the  instant.  Since  the 
equation  of  time  is  given  in  the  Nautical  Almanac  for  Greenwich 
noon  its  value  at  the  desired  instant  must  be  found  by  adding 
or  subtracting  the  increase  or  decrease  since  Greenwich  noon 


44  PRACTICAL  ASTRONOMY 

Example.  Find  the  local  apparent  time  at  Boston  at  2  P.M.  (local  mean  time) 
Oct.  28,  1910.  The  Greenwich  Mean  Time  corresponding  to  2  P.M.  local  mean 
time  is  6h  44™  i8s  P.M.  The  equation  of  time  at  G.  M.  N.  Oct.  28,  1910,  is 
i6m  04s. 29  (to  be  added  to  mean  time);  the  hourly  increase  is  bs.2o8.  The  correc- 
tion to  the  equation  of  time  is  6h.j4X  os.2o8  =  is.4o.  The  equation  of  time  at 
2  P.M.  is  therefore  i6mo$s.6g. 

L.  M.  T.  =  2h  oom  oos. oo 

Equa.  of  T.  =       16    05  .69 

L.  A.  T.  =  2h  i6m  05^69 

29.   Astronomical  and  Civil  Time. 

For  ordinary  purposes  it'  is  found  convenient  to  divide  the 
solar  day  into  two  parts  of  i2h  each;  from  midnight  to  noon  is 
called  A.M.  (ante  meridiem),  and  from  noon  to  midnight  is 
called  P.M.  (post  meridiem).  The  date  changes  at  the  instant 
of  midnight.  This  mode  of  reckoning  time  is  called  Civil  Reck- 
oning. In  astronomical  work  this  subdivision  of  the  day  is  not 
convenient.  For  simplicity  in  calculation  the  day  is  divided 
into  24h,  numbered  consecutively  from  oh  to  24*.  As  it  is  not 
convenient  to  have  the  date  change  during  the  night,  the  astro- 
nomical date  begins  at  noon  or  oh.  This  is  called  Astronomical 
Time.  In  using  the  Nautical  Almanac  it  should  be  remembered 
that  it  is  necessary  to  change  the  date  and  hours  to  astronomical 
time  before  taking  out  the  desired  data.  In  order  to  change 
from  one  kind  of  time  to  the  other  it  is  only  necessary  to  remem- 
ber that  the  astronomical  day  begins  at  noon  of  the  civil  day  of 
the  same  date;  that  is,  in  the  afternoon  the  dates  and  the  hours 
will  be  the  same,  but  in  the  forenoon  the  astronomical  date  is 
one  day  less  and  the  hours  are  12  greater  than  in  the  civil  time. 

Examples. 

Astr.  Time  May  10,  15*  =  Civil  Time  May  n,  3*  A.M. 
Jan.  3,    7*  =      "        "      Jan.   3,  7*  P.M. 

From  these  examples  the  following  rules  may  be  derived: 
To  change  Civil  Time  to  Astronomical  Time, 
If  A.M.,  add  i2h  and  drop  i  day  from  date,  and  drop  the  A.M. 
If  P.M.,  drop  the  P.M. 


MEASUREMENT   OF   TIME  45 

To  change  Astronomical  Time  to  Civil  Time* 

If  less  than  i2h,  mark  it  P.M. 

If  greater  than  i2h,  subtract  i2\  add  i  day  to  date,  and  mark 
it  A.M. 

30.   Relation  between  Longitude  and  Time. 

The  hour  angle  of  the  sun  at  any  given  meridian  at  a  given 
instant  is  the  local  solar  time  at  that  meridian,  and  will  be 
apparent  or  mean  time  according  as  the  true  sun  or  the  mean 
sun  is  considered.  The  hour  angle  of  the  sun  at  Greenwich  at 
the  same  instant  is  the  corresponding  Greenwich  solar  time. 
The  difference  between  the  two  hour  angles  is  the  longitude 
of  the  place  from  Greenwich,  expressed  either  in  degrees  or  in 
hours  according  as  the  hour  angles  themselves  are  expressed 
in  degrees  or  in  hours.  Similarly  the  difference  in  local  solar 
time  of  any  two  places  at  a  given  instant  is  their  difference  in 
longitude  in  hours,  minutes,  and  seconds.  In  Fig.  32,  AC  is 
the  hour  angle  of  the  sun  at  Greenwich  (G),  or  the  Greenwich 
solar  time.  BC  is  the  hour  angle  of  the  sun  at  the  meridian 
through  P,  or  the  local  solar  time  at  P.  The  difference,  AB, 
is  the  longitude  of  P  west  of  Greenwich.  It  should  be  observed 
that  the  reasoning  is  exactly  the  same  whether  C  represents  the 
true  sun  or  the  fictitious  sun.  The  same  result  would  also  be 
found  if  the  point  C  were  to  represent  the  vernal  equinox.  The 
arc  AC  would  then  be  the  hour  angle  of  the  equinox,  i.e.,  the 
Greenwich  Sidereal  Time.  BC  would  be  the  Local  Sidereal 
Time,  and  AB  the  difference  in  longitude.  The  measurement 
of  longitude  is  therefore  independent  of  the  kind  of  time  used, 
because  in  each  case  the  angular  distances  to  A  and  B  are  meas- 
ured from  the  same  point  C  on  the  equator,  and  the  difference  in 
these  angles  does  not  depend  upon  the  position  of  this  point 
nor  upon  the  speed  with  which  this  point  has  moved  up  to  the 
position  at  C. 


*  The  student  may  find  it  helpful  to  plot  the  time  along  a  straight  line,  and  to 
write  two  sets  of  numbers,  one  for  Civil  Dates  and  the  other  for  Astronomical  Dates. 


46 


PRACTICAL  ASTRONOMY 


The  difference  in  the  sidereal  times  at  meridian  A  and  meridian 
B  (Fig.  32)  is  the  interval  of  sidereal  time  during  which  a  star 
would  go  from  A  to  B.  Since  the  star  requires  24  sidereal  hours 
to  travel  from  meridian  A  to  meridian  A  again,  the  time  interval 
from  A  to  B  bears  the  same  relation  to  24*  that  the  longitude 


Pole 


FIG.  32 

difference  bears  to  360°.  The  difference  in  the  mean  solar  times 
at  A  and  B  is  the  number  of  mean  solar  hours  that  the  sun 
would  take  to  go  from  A  to  B,  and  since  the  sun  takes  24  solar 
hours  to  go  from  A  to  A  again,  the  time  interval  from  A  to  B 
bears  the  same  ratio  to  24  solar  hours  as  when  sidereal  time  was 
used.  The  difference  in  longitude  is  therefore  correctly  given 
when  either  sidereal  or  solar  times  are  comparedo 

The  method  of  changing  from  Greenwich  to  local  time  and  the 
reverse  is  illustrated  by  the  following  examples. 

Example  i.   The  Greenwich  astronomical  time  is  jh  40™  ios.o.     Required  the 
local  time  at  a  meridian  4^  50 m  2is.o  West. 

G.  M.  T.  =  7h  4om  ios.o 
Long.  West  =  4    50    21  .o 

L.  M.  T.  =  2h  49W  49s.o     (P.M.) 


MEASUREMENT  OF  TIME  47 

Example   2.   The  Greenwich  mean  time  is  3^  20™  i6s.5.     Required  the   local 
mean  time  at  a  place  whose  longitude  is  120°  10'  West. 

G.  M.  T.  +  24h  =  2jh  2om  i6s.s 

Long.  West  =    &h  oom  4o*.o 

L.  M.  T.  =  igh  igm  36^5 

=    7h  igm  36s.s  A.M. 

Example  3.   The  mean  time  at  a  place  3^  East  longitude  is  ioh  A.M.     Required 
the  Greenwich  mean  time. 

L.  M.  T.  =  22h  oom  oos. o 

Long.  East  =    3^  oom  oos.o 

G.  M.  T.  =  igh  oom  oos.o 

=      7A00™OOs.O  A.M. 

Since  a  circle  may  be  divided  either  into  24^  or  into  360°,  the 
relation  between  these  two  units  is  constant.     From  the  fact  that 


24"  =  36o° 

we  have  also  ih=  15°, 

i~  =  15' 

The  following  equivalents  are  also  convenient: 


By  means  of  these  two  sets  of  equivalents  time  may  be  con- 
verted into  degrees,  or  the  contrary,  without  writing  down  the 
intermediate  steps.  In  the  following  examples  the  intermediate 
steps  are  written  down  in  order  to  show  the  process  followed. 

Example  i.   Convert  6^  35m  5is  into  degrees. 

•  6h  =  90° 
35m  =  32m  +  3m=    8°  45' 

Total  =  98°  5  7'  45" 

Example  2.   Convert  47°  17' 35"  into  hours. 

47°  =  45°  +  2°    =  3h  08™ 

!/   =    I5'  -f  2'      =          OimoSS 

35"  =  30"  +  5"  = 02.33 

Total  = 


48 


PRACTICAL  ASTRONOMY 


It  should  be  observed  that  the  relation  15°  =  ih  is  quite 
independent  of  the  length  of  time  that  has  elapsed.  A  star 
takes  one  sidereal  hour  to  move  over  15°  of  hour  angle;  the  sun 
takes  one  solar  hour  to  move  over  15°  of  hour  angle.  In  the 
sense  in  which  it  is  used  here,  ih  means  an  angle,  and  not  an 
absolute  interval  of  time. 

31.  Relation  between  Sidereal  Time,  Right  Ascension,  and 
Hour  Angle  of  any  Point  at  a  Given  Instant. 

In  Fig.  33  the  hour  angle  of  the  equinox,  or  local  sidereal  time 
at  the  meridian  through  P,  is  the  arc  A  V.  The  hour  angle  of 


Pole 


FIG.  33 

the  star  5  at  the  meridian  through  P  is  the  arc  AB.  The  right 
ascension  of  the  star  5  is  the  arc  VB.  It  is  evident  from  the 
figure  that 

AV  =  VB  +  AB, 
or  S  =  R  +  P,  [37] 

where  R  =  the  right  ascension  and  P  =  the  hour  angle  of  the 
point  5,  and  5  =  the  sidereal  time;  or,  in  words, 

Sidereal  Time  =  Right  Ascension  -\-Hour  Angle.        [38] 


MEASUREMENT   OF   TIME  49 

This  relation  is  a  perfectly  general  one  and  will  be  found  to  hold 
true  for  all  points  on  the  sphere,  provided  it  is  agreed  to  reckon 
the  sidereal  time  beyond  24*  when  necessary.  For  example,  if 
the  hour  angle  is  ioh  and  the  right  ascension  is  20*,  the  resulting 
sidereal  time  is  30*.  This  means  that  the  equinox  has  made  a 
complete  revolution  and  has  gone  6h,  or  90°,  on  the  next  revolu- 
tion; the  actual  reading  of  the  sidereal  clock  would  be  6h.  In 
the  reverse  case,  when  it  is  necessary  to  subtract  2oh  from  6* 
to  obtain  the  hour  angle,  the  6h  must  first  be  increased  by  24^ 
and  the  right  ascension  subtracted  from  the  sum  to  obtain  the 
hour  angle,  ioh. 

32.  Star  on  the  Meridian. 

At  the  instant  when  the  star  is  on  the  meridian  its  hour  angle 
is  oh  and  the  equation  becomes 

Sidereal  Time  =  Right  Ascension;  [39] 

that  is,  the  right  ascension  of  a  star  equals  the  local  sidereal  time 
at  which  that  star  crosses  the  meridian.  (See  Art.  20,  p.  36.) 

33.  Relation  between  Mean  Solar  and  Sidereal  Intervals  of 
Time. 

It  has  already  been  stated  that  on  account  of  the  earth's 
orbital  motion  the  sun  has  an  apparent  eastward  motion  among 
the  stars  of  nearly  i°  per  day.  This  eastward  movement  of  the 
sun  makes  the  intervals  between  the  sun's  transits  greater  by 
nearly  4™  than  the  intervals  between  the  transits  of  the  equinox, 
that  is,  the  solar  day  is  nearly  4™  longer  than  the  sidereal  day. 
In  Fig.  34  let  C  and  C'  be  the  positions  of  the  earth  on  two 
consecutive  days.  When  the  observer  is  at  O  it  is  local  noon. 
After  the  earth  makes  one  complete  rotation,  the  observer  will 
be  at  O',  and  the  sidereal  time  will  be  exactly  the  same  as  it  was 
the  day  before  when  he  was  at  O.  But  the  sun's  direction  is 
now  CO",  so  the  earth  must  turn  through  the  angle  O'C'O" 
in  order  to  bring  the  sun  again  on  the  observer's  meridian. 
Since  this  angle  is  about  i°  it  takes  about  4m  longer  to  complete 
the  solar  day  than  it  does  to  complete  the  sidereal  day.  Since 


50  PRACTICAL  ASTRONOMY 

each  kind  of  day  is  subdivided  into  hours,  minutes,  and  seconds, 
all  of  these  units  in  solar  time  will  be  proportionally  longer  than 
the  corresponding  units  of  sidereal  time.  If  two  clocks,  one 
regulated  to  mean  solar  time  and  the  other  to  sidereal  time,  were 
started  at  the  same  instant,  both  reading  oh,  the  sidereal  clock 
would  immediately  begin  to  gain  on  the  solar  clock,  the  gain 


FIG.  34 

being  exactly  proportional  to  the  time  interval,  that  is,  about  10* 
per  hour,  or  more  nearly  3™  56*  per  day. 

In  order  to  find  the  exact  relation  between  the  two  kinds  of 
time  it  should  be  observed  that  the  number  of  sidereal  days  in 
the  year  is  exactly  one  greater  than  the  number  of  solar  days, 
because  the  sun  comes  back  to  the  equinox  at  the  end  of  one 
year.  The  length  of  the  tropical*  year  is  found  to  be  365.2422 

*  The  tropical  year  is  the  interval  of  time  between  two  successive  passages  of 
the  sun  over  the  vernal  equinox.  The  sidereal  year  is  the  interval  between  two 
passages  of  the  sun  across  the  hour  circle  through  a  fixed  star  on  the  equator.  On 
account  of  the  movement  of  the  equinox  caused  by  precession,  the  tropical  year  is 
about  2om  shorter  than  the  sidereal  year. 


MEASUREMENT   OF  TIME  51 

mean  solar  days.  The  relation  between  the  two  kinds  of  day  is 
therefore 

366.2422  sidereal  days  =  365.2422  solar  days,  [40] 

or  i  sidereal  day  =  0.99726957  solar  day,  [41] 

and  i  solar  day  =  1.00273791  sidereal  days.      [42] 

Equations  [41]  and  [42]  may  be  written 

24^  sidereal  time  =  (24^  —  3™  55^909)  mean  solar  time, 
24^  mean  solar  time  =  (24^  +  3m  56^.555)  sidereal  time. 

These  equations  may  be  put  in  more  convenient  form  for  com- 
putation by  expressing  the  difference  in  time  as  a  correction 
to  be  applied  to  any  interval  of  time  to  change  it  from  one  kind 
of  unit  to  the  other.  If  Im  is  a  mean  solar  interval  and  7S  the 
corresponding  number  of  sidereal  units,  then 

Is    =   Im  +   .00273791    X   Im  [43] 

and  Im  =  I8    -~  .00273043  X  Is.  [44] 

Tables  II  and  III  are  constructed  by  multiplying  different  values 
of  Im  and  Is  by  these  constants.  More  extended  tables  may  be 
found  in  the  Nautical  Almanac.  The  use  of  Tables  II  and  III 
is  illustrated  by  the  following  examples. 

Examples. 

Reduce  9^  23  :l  5is.oof  sidereal  time  to  the  equivalent  number 
of  solar  units.  From  Table  II,  opposite  gh  is  the  correction 
—  im  28*466;  opposite  23™  in  the  4th  column  is  — 3^.768;  and 
opposite  5is  in  the  last  column  is  os.i39.  The  sum  of  these 
three  partial  corrections  is  —  im  32s. 373,  which  is  the  amount  to 
be  subtracted  from  gh  23™  5is.o  to  reduce  it  to  the  equivalent 
solar  interval,  gh  22™  i8s.62j. 

Reduce  jh  iom  solar  time  to  sidereal  time.  The  correction  for 
7*,  Table  III,  is  +  im  o8s.995,  and  for  iom  is  1^.643.  The  sum, 
iw  10^.638,  added  to  7^  iow  gives  7*  n™  ios.638  of  sidereal  time. 

This  reduction  may  be  made  approximately  by  the  following 
rule:  the  correction  equals  ios  per  hour  diminished  by  is  for 


S2  PRACTICAL  ASTRONOMY 

every  6h  in  the  interval.  The  correction  for  6h  would  be 
6  X  ios  —  is  =  59s.  This  rule  is  based  on  a  change  of  3™ 
56s  per  day.  For  changing  solar  into  sidereal  the  error  is 
os.023  per  hour;  for  sidereal  into  solar  the  error  is  0^.004  per 
hour. 

It  should  be  kept  in  mind  that  the  conversion  of  time  discussed 
in  this  article  concerns  the  change  from  one  kind  of  unit  to 
another,  like  changing  from  yards  to  metres,  and  is  not  the  same 
as  changing  from  the  local  sidereal  time  to  the  local  solar  time 
at  a  particular  instant. 

34.  Relation  between  Sidereal  Time  and  Mean  Solar  Time 
at  any  Instant. 

If  in  Fig.  33,  Art.  31,  the  point  B  is  taken  to  represent  the 
mean  sun,  then  equation  [37]  becomes 

S  =  Rs  +  P.,  [45] 

where  Rs  and  Ps  are  the  right  ascension  and  the  hour  angle  of 
the  mean  sun  at  the  instant  considered.  Ps  is  the  local  mefan 
time  by  the  definition  given  in  Art.  26.  If  the  equation  is 
written 

S  -  Ps  =  R9,  [46] 

then,  since  the  value  of  the  right  ascension  Rs  does  not  depend 
upon  the  time  at  any  particular  meridian,  but  only  upon  the 
absolute  instant  of  time  considered,  it  is  evident  that  the  differ- 
ence between  sidereal  time  and  mean  time  at  any  instant  is  the 
same  for  all  places  on  the  earth.  The  actual  values  of  S  and 
Ps  will  of  course  be  different  at  different  meridians,  but  the 
difference  between  the  two  is  a  constant  for  all  places  for  the 
given  instant.  In  order  that  Equa.  [45]  shall  hold  true  it  is 
essential  that  Rs  and  Ps  shall  refer  to  the  same  position  of  the 
sun,  that  is,  to  the  same  absolute  instant  of  time.  The  right 
ascension  of  the  sun  obtained  from  the  Nautical  Almanac  is  its 
value  at  the  instant  of  the  Greenwich  Mean  Noon  preceding, 
that  is,  at  the  beginning  of  the  astronomical  day  at  Greenwich.* 

*  The  dates  are  always  in  mean  solar  days,  not  in  sidereal  days. 


MEASUREMENT   OF   TIME 


53 


To  reduce  this  right  ascension  to  its  value  at  the  desired  instant 
it  is  necessary  to  multiply  the  hourly  increase  in  the  right  ascen- 
sion of  the  mean  sun  by  the  number  of  solar  hours  elapsed  since 
the  instant  of  Greenwich  Mean  Noon.  The  hourly  increase  in 
the  right  ascension  of  the  mean  sun  is  constant  and  is  evidently 
equal  to  the  correction  in  Table  III,  for  the  difference  between 
sidereal  and  solar  time  is  caused  by  the  sun's  motion,  and  the 
amount  of  the  difference  for  any  number  of  hours  is  exactly 
equal  to  the  increase  in  the  right  ascension.  If  it  is  desired  to 
find  the  increase  for  any  number  of  solar  hours,  Table  III  should 
be  used;  for  sidereal  hours  use  Table  II.  Equation  [45]  may 
be  written 

S  =  Rs  +  Ps  +  C,  [47] 

where  Rs  refers  to  the  instant  of  the  preceding  local  mean  noon, 
and  C  is  the  correction  (Table  III)  to  reduce  Ps  to  a  sidereal 
interval,  or  to  reduce  Rs  to  its  value  at  the  time  P8. 

In  Fig.  35  suppose  that  the  sun  51  and  a  star  S'  passed  the 
meridian  M  at  the  same  instant,  and  at  the  mean  time  Pa  it  is 

M 


FIG.  35 

desired  to  compute  the  sidereal  time.     Since  the  sun  is  moving 
at  a  slower  rate  than  the  star,  it  will  describe  the  arc  MS  (  =  Ps) 


54  PRACTICAL  ASTRONOMY 

while  the  star  moves  from  M  to  S'.  The  arc  SSf,  or  C,  repre- 
sents the  gain  of  sidereal  on  mean  time  in  the  mean  time  interval 
MS  or  Ps.  But  S'  is  the  position  of  the  sun  at  noon,  so  that 
VSf  is  the  sun's  right  ascension  at  the  preceding  mean  noon,  or 
Rs.  The  right  ascension  desired  is  FS,  so  Rs  must  evidently 
be  increased  by  the  arc  SS',  or  C. 

If  it  is  desired  to  find  the  mean  solar  time  corresponding  to 
a  given  instant  of  local  sidereal  time,  the  equation  is 

Sidereal  interval  from  noon  =  S  —  Rs,  [48] 

or  Mean  time  =  Ps  =  S  —  Rs  —  C',  [49] 

where  C'  is  the  correction  from  Table  II  to  reduce  S  —  Rs  to  a 
solar  interval,  and  represents  the  increase  in  the  sun's  right 
ascension  in  S  —  Rs  sidereal  hours. 
Examples. 

To  find  the  Greenwich  Sidereal  Time  corresponding  to  Greenwich  Mean  Time 
<)h  22m  1 8s.  60  on  Jan.  7,  1907.  The  right  ascension  of  the  mean  sun  at  Greenwich 
Mean  Noon  is  found  from  the  Nautical  Almanac  to  "be  I9*o3m36*.38.  The  cor- 
rection to  reduce  gh  22™  i8s.6o  to  sidereal  time  (Table  III)  is  +im  32^.37.  Then, 
applying  Equa.  [47], 

Rs  =  19*  o3TO  36S.38 

Ps  =    9    22    18  .60 

C=  *    3^37 

5  =  28^27^27^.35 

Sidereal  Time  =    4^  27™  27^35 

To  find  the  Greenwich  Mean  Solar  Time  when  the  Greenwich  Sidereal  Time  is 
4*  27m  27^.35  on  Jan.  7,  1907. 

5  =    28^  27m  27S.3S 

Rs  =  19   03    36  .38 

S  —  Rs  =  9    23    50  .97 

C  (Table  II)  =  -i    32.37 

Mean  Time  =  gh  22™  i8s.6o 

If  the  change  from  sidereal  to  solar  time  (or  vice  versa)  is  to  be 
made  at  any  meridian  other  than  Greenwich,  the  right  ascension 
of  the  sun  for  local  noon  must  be  found  by  multiplying  the 
increase  per  solar  hour  by  the  number  of  solar  hours  since  Green- 
wich noon,  that  is,  by  the  number  of  hours  in  the  longitude,  and 


MEASUREMENT  OF  TIME  55 

adding  this  to  the  value  of  Rs  from  the  Almanac  if  the  place  is 
west  of  Greenwich,  subtracting  if  east.*  The  correction  may 
be  taken  from  Table  III.  If  the  sidereal  time  in  the  above 
example  is  assumed  to  be  the  time  at  a  meridian  5*  (75°)  west  of 
Greenwich,  the  computation  would  be  modified  as  follows: 

R8'  =  iQh  o3m  36S.38 

Correction  for  5^  longitude  =  49  .28 

Rs  —  19    04    25  .66 

5  =  28    27    27  .35 

S  —  Rs  =    9    23    01  .69 

C  =  i    32  .24 


Local  Mean  Time  =    gh  21™  29*. 45 

It  is  evident  that  at  the  instant  of  mean  noon  Ps  =  o  and 
Rs  =  S.  At  mean  noon,  therefore,  the  sidereal  time  equals 
the  right  ascension  of  the  mean  sun.  This  quantity  will  be 
found  in  the  Almanac  under  both  headings,  "  Sidereal  Time  of 
Mean  Noon"  and  "  Right  Ascension  of  the  Mean  Sun."  (See 
P-  65.) 

The  reduction  of  mean  solar  time  to  sidereal  time,  or  the  re- 
verse, may  be  made  also  by  first  changing  the  given  local  time 
to  the  corresponding  instant  of  Greenwich  time,  then  making 
the  transformation  as  before,  and  finally  changing  back  to  the 
meridian  of  the  place.  Take,  for  example,  the  case  just  worked 
out. 

Local  Sidereal  Time  =  28^  2jm2js.^5 

Longitude  =    5    oo  oo 

Greenwich  Sidereal  Time  =  33    27   27  .35 

Rs  at  Gr.  M.  Noon  =  19    03  36  .38 

Sidereal  Interval  from  Noon  =  14    23   50  .97 

C'  =       —2  21  .52 

Greenwich  Mean  Time  =  14    21   29  .45 

Longitude  =    5    oo  oo 
Local  Mean  Time  =    9^  2im29s.45 

The  result  agrees  with  that  obtained  by  the  former  method. 
This  method  is  quite  as  simple  as  the  preceding,  especially  when 

*  It  should  be  remembered  that  the  sun's  R.  A.  is  always  increasing. 


56  PRACTICAL  ASTRONOMY 

Standard  Time  is  to  be  computed,  for  the  final  correction  will 
always  be  a  whole  number  of  hours.  Care  should  be  taken 
always  to  use  the  right  ascension  of  the  sun  at  the  noon  preced- 
ing the  given  time.  Suppose  that  the  instant  of  ioh  A.M.  May  5 
is  to  be  converted  into  sidereal  time,  the  longitude  of  the  place 
being  4h  44™  i8s  west.  Civil  time  ioh  A.M.  May  5  =  Astr. 
time  22h  May  4.  If  the  first  method  is  followed,  the  right 
ascension  of  the  sun  employed  should  be  that  of  noon  May  4. 
If  the  reduction  is  made  by  first  changing  to  Greenwich  time, 
then  22h  +  4h  44m  i8s  =  26h  44m  i8s  May  4  =  ^  44™  i8s 
May  5.  The  right  ascension  for  the  latter  case  would  be  that 
for  noon  of  May  5. 

35.   Standard  Time. 

From  the  definition  of  mean  solar  time  it  will  be  seen  that  at 
any  given  instant  the  solar  times  at  two  places  will  differ  from 
each  other  by  an  amount  depending  upon  the  difference  in  the 
longitudes.  All  places  will  have  different  local  times  except 
where  they  happen  to  be  on  the  same  meridian.  Previous  to  the 
year  1883  it  was  customary  in  this  country  for  each  large  city 
or  town  to  use  the  mean  time  at  its  own  meridian,  and  for  all 
other  places  in  the  vicinity  to  adopt  the  same  time.  Before 
railroad  travel  became  extensive  this  change  of  time  from  one 
point  to  another  caused  no  great  difficulty,  but  with  the  in- 
creased amount  of  railroad  and  telegraph  business  these  frequent 
and  irregular  changes  in  time  became  so  inconvenient  that  in 
1883  a  uniform  system  of  time  was  adopted  in  the  United  States. 
The  country  is  divided  into  time  belts  each  theoretically  15°  in 
width;  these  are  known  as  the  Eastern,  Central,  Mountain  and 
Pacific  time  belts,  and  places  in  these  belts  use  the  mean  local 
time  of  the  75°,  90°,  105°  and  120°  meridians  respectively.  The 
time  at  the  60°  meridian  is  called  Atlantic  time  and  is  used  in 
the  Eastern  Provinces  of  Canada.  The  actual  positions  of  the 
dividing  lines  between  these  belts  depend  upon  the  positions  of 
the  principal  cities  and  the  railroads  (see  Fig.  36),  but  the  change 
of  time  from  one  belt  to  another  is  always  exactly  one  hour.  The 


MEASUREMENT  OF  TIME 


57 


5  8  PRACTICAL  ASTRONOMY 

minutes  and  seconds  of  all  clocks  are  the  same  as  the  minutes 
and  seconds  of  the  Greenwich  clock.  When  it  is  noon  at  Green- 
wich it  is  8  A.M.  Atlantic  time,  7  A.M.  Eastern  time,  6  A.M. 
Central  time,  5  A.M.  Mountain  time,  and  4  A.M.  Pacific  time. 

The  change  from  local  to  standard  time,  or  the  contrary,  con- 
sists in  expressing  the  difference  in  longitude  between  the  local 
meridian  and  the  standard  meridian  in  units  of  time,  and  adding 
or  subtracting  this  correction,  remembering  that  the  farther 
west  a  place  is,  the  earlier  it  is  in  the  day  at  any  given  instant 
of  time. 

Examples. 

Find  the  standard  time  at  a  place  71°  west  of  Greenwich  when 
the  local  time  is  4h  20™  oos  P.M.  In  longitude  71°  the  standard 
time  would  be  that  of  the  75°  meridian.  The  difference  in 
longitude  is  4°  =  i6w  .  Since  the  standard  meridian  is  west  of 
the  71°  meridian,  the  time  is  i6w  earlier  than  the  local  time.  The 
standard  time  is  therefore  4*  04™  oos  P.M. 

Find  the  local  time  at  a  place  91°  west  of  Greenwich  when  the 
Central  time  is  gh  oom  oo8  A.M.  The  difference  in  longitude  is 
i°  =  4™.  Since  the  place  is  west  of  the  standard  meridian,  the 
time  is  earlier.  The  local  time  is  therefore  Sh  56™  oos  A.M. 

Standard  time  is  used  not  only  in  the  United  States  but  in  a 
majority  of  the  countries  of  the  world;  in  nearly  all  cases  these 
systems  of  standard  time  are  based  on  the  meridian  of  Green- 
wich as  the  prime  meridian.  Germany,  for  example,  uses  the 
local  mean  time  at  the  meridian  ih  east  of  Greenwich;  Japan 
uses  that  of  the  meridian  gh  east  of  Greenwich;  Turkey,  2h  east 
of  Greenwich,  etc. 

36.  The  Date  Line. 

If  a  person  were  to  start  at  Greenwich  at  the  instant  of  noon 
and  travel  westward  rapidly  enough  to  keep  the  sun  always  on 
his  meridian  he  would  get  back  to  Greenwich  24^  later,  but  his 
own  (local)  time  would  not  have  changed  but  would  have 
remained  noon  all  the  time.  In  travelling  westward  at  a  slower 
rate  the  same  thing  occurs,  only  in  a  longer  interval  of  time. 


MEASUREMENT   OF  TIME  59 

The  traveller  has  to  set  his  watch  back  every  day  in  order  to 
keep  it  regulated  to  the  meridian  at  which  his  noon  occurs.  As 
a  consequence,  his  watch  has  recorded  one  day  less  than  it  has 
actually  run,  and  his  calendar  is  one  day  behind  that  of  a  person 
who  remains  at  Greenwich.  If  the  traveller  goes  east  he  has  to 
set  his  watch  ahead  every  day,  and  after  circumnavigating  the 
globe  his  calendar  is  one  day  ahead  of  what  it  should  be.  In 
order  that  the  calendar  may  be  everywhere  uniform,  it  is  agreed 
to  change  the  date  at  the  meridian  180°  from  Greenwich.  When- 
ever a  ship  crosses  the  180°  meridian  going  westward,  a  day  is 
omitted  from  the  calendar,  and  when  going  eastward  a  day  is 
repeated.  In  practice  the  change  is  made  at  midnight  near  the 
1 80°  meridian,  not  at  the  instant  of  crossing.  The  date  line 
actually  used  does  not  follow  the  180°  meridian  in  all  places,  but 
is  deflected  so  as  not  to  separate  the  Aleutian  islands,  and  in 
the  South  Pacific  ocean  it  passes  east  of  several  groups  of  islands 
so  as  not  to  change  the  date  formerly  used  in  these  islands. 
37.  The  Calendar. 

Previous  to  the  time  of  Julius  Caesar  the  calendar  was  based 
upon  the  lunar  month,  and,  as  this  resulted  in  a  continual  change 
in  the  date  at  which  the  seasons  occurred,  the  calendar  was  fre- 
quently changed  in  an  arbitrary  manner  in  order  to.  keep  the 
seasons  in  their  places,  the  result  being  extreme  confusion  in  the 
dates.  In  the  year  45  B.C.  Julius  Caesar  reformed  the  calendar 
and  introduced  one  based  upon  a  year  of  365^  days,  since  called 
the  Julian  calendar.  The  J  day  was  taken  care  of  by  making 
the  year  contain  365  days,  except  every  4th  year,  called  leap 
year,  which  contained  366;  the  extra  day  was  added  to  February 
in  such  years  as  were  divisible  by  4.  The  year  was  begun  on 
Jan.  i ;  previously  it  had  begun  in  March.  Since  the  year  con- 
tains actually  365**  5^  48 m  46*,  this  difference  of  nm  14*  caused 
a  gradual  change  in  the  dates  at  which  the  seasons  occurred. 
After  many  centuries  the  difference  had  accumulated  to  about 
10  days,  so  in  1582  Pope  Gregory  XIII  ordered  that  the  calendar 
should  be  corrected  by  dropping  ten  days  and  that  future  dates 


6o 


PRACTICAL  ASTRONOMY 


should  be  computed  by  omitting  the  366th  day  in  those  leap  years 
which  occurred  in  century  years  not  divisible  by  400;  that  is, 
such  years  as  1700,  1800  and  1900  should  not  be  counted  as  leap 
years.  This  is  the  calendar  used  at  the  present  time. 

The  dates  of  the  actual  adoption  of  these  changes  were  quite  different  in  differ- 
ent countries.  The  Gregorian  Calendar  was  adopted  in  1582  by  the  Catholic 
nations.  It  was  not  until  1752,  however,  that  these  changes  were  made  in  Eng- 
land. Up  to  that  time  the  legal  year  had  begun  on  March  25,  and  the  dates  were 
reckoned  according  to  the  Julian  Calendar.  It  is  necessary  therefore  when  con- 
sulting records  of  this  period  to  determine  whether  they  are  dated  according  to 
the  "  Old  Style  "  or  the  "  New  Style,"  as  the  two  are  called. 

In  Russia  and  in  the  Greek  Church  the  Julian  Calendar  is  still  in  use  in  its 
original  form. 

Questions  and  Problems 

1.  (a)  Prove  by  direct  computation  of  sidereal  time  from  Fig.  37  that 

R  +  P  =  24h  +  S, 

in  which  R  and  P  are  the  right  ascension  and  hour  angle  of  the  star  S,  and  5  is  the 
sidereal  time,  or  hour  angle  of  V. 

(b)  Prove  the  same  relation  when  V  is  at  the  point   V.     (See  Art.  31, 
P- 48.) 

2.  Prove  that  the  difference  in  longitude  of  two  points  is  independent  of  the 
kind  of  time  used,  by  selecting  two  points  at  which  the  solar  time  differs  by  say 
3^,  and  then  converting  the  solar  time  at  each  place  into  sidereal  time. 


FIG.  37 


FIG.  38 


3.  Make  a  design  for  a  horizontal  sun  dial  for  a  place  whose  latitude  is  42°  21'  N. 
The  gnomon  ad  (Fig.  38),  or  line  which  casts  the  shadow  on  the  horizontal  plane, 
must  be  parallel  to  the  earth's  rotation  axis;  the  angle  which  the  gnomon  makes 


MEASUREMENT   OF  TIME  6 1 

with  the  horizontal  plane  therefore  equals  the  latitude.  The  shadow  lines  for  the 
hours  (X,  XI,  XII,  I,  II,  etc.)  are  found  by  passing  planes  through  the  gnomon 
and  finding  where  they  cut  the  horizontal  plane  of  the  dial.  The  vertical  plane 
adb  coincides  with  the  meridian  and  therefore  is  the  noon  (XII^)  line.  The  other 
planes  make,  with  the  vertical  plane,  angles  equal  to  some  multiple  of  15°.  In 
finding  the  trace  dc  of  one  of  these  planes  on  the  dial  it  should  be  observed  that  the 
foot  of  the  gnomon,  d,  is  a  point  common  to  all  such  traces.  In  order  to  find  another 
point  c  on  any  trace,  or  shadow  line,  pass  a  plane  abc  through  some  point  a  on  the 
gnomon  and  perpendicular  to  it.  This  plane  (the  plane  of  the  equator)  will  cut 
an  east  and  west  line  ce  on  the  dial.  If  a  line  be  drawn  in  this  plane  making  an 
angle  of  n  X  15°  with  the  meridian  plane,  it  will  cut  ce  at  a  point  c  which  is  on  the 
shadow  line.  Joining  c  with  the  foot  of  the  gnomon  gives  the  required  line. 

In  making  a  design  for  a  sun  dial  it  must  be  remembered  that  the  west  edge  of 
the  gnomon  casts  the  shadow  in  the  forenoon  and  the  east  edge  in  the  afternoon; 
there  will  be  of  course  two  noon  lines,  and  the  two  halves  of  the  diagram  will  be 
symmetrical  and  separated  from  each  other  by  the  thickness  of  the  gnomon.  The 
dial  may  be  placed  in  position  by  levelling  the  horizontal  surface  and  then  com- 
puting the  watch  time  of  apparent  noon  and  turning  the  dial  so  that  the  shadow  is 
on  the  XII^  line  at  the  calculated  time. 

Prove  that  the  horizontal  angle  bdc  is  given  by  the  relation 

tan  bdc  =  tan  P  sin  L, 
in  which  P  is  the  sun's  hour  angle  and  L  is  the  latitude. 

4.  Why  are  the  sun's  and  moon's  right  ascension  always  increasing? 

5.  The  Local  Apparent  Time  at  a  point  A,  in  longitude  95°  W.,  is  IOA  30™  A.M., 
Jan.  i,  1917.     If  the  equation  of  time  at  G.  M.  N.  is  —  3™  34S.47  and  the  hourly 
increase  is  is.i84,  what  is  the  Mean  (civil)  Time  at  A?    What  is  the  Astronomical 
Mean  Time  at  A?    What  is  the  Greenwich  Mean  Time?    The  Central  Standard 
Time?     What  is  the  Local  Mean  Time  at  the  same  instant  at  a  point  B  in  longi- 
tude 110°  W.?     If  the  right  ascension  of  the  mean  sun  at  G.  M.  N.,  Jan.  i,  1917,  is 
1 8*  42™  1 5s. 64,  what  is  the  Greenwich  Sidereal  Time?     What  is  the  Local  Sidereal 
Time  at  A? 


CHAPTER  VI 

THE  AMERICAN   EPHEMERIS   AND   NAUTICAL 
ALMANAC  —  STAR    CATALOGUES  —  INTERPOLATION 

38.    The  Ephemeris. 

In  discussing  the  problems  of  the  previous  chapters  it  has  been 
assumed  that  the  right  ascensions  and  declinations  of  the  celes- 
tial objects  and  various  other  data  mentioned  are  known  to  the 
computer.  These  data  consist  of  results  calculated  from  obser- 
vations made  with  large  instruments  at  the  astronomical  obser- 
vatories, and  are  published  by  the  Government  in  the  American 
Ephemeris  and  Nautical  Almanac.*  This  publication  contains 
the  declinations  and  right  ascensions  of  the  sun,  moon,  planets 
and  stars,  as  well  as  the  angular  semidiameters,  horizontal  paral- 
laxes, the  equation  of  time  and  other  necessary  data.  The 
Almanac  is  published  by  the  Navy  Department,  Washington, 
D.  C.,  and  may  be  obtained,  through  the  regular  agents,  about 
two  years  in  advance  of  the  date  for  which  it  is  computed. 

It  should  be  observed  that  the  computed  quantities  given  in 
the  almanac  all  vary  with  the  time  and  are  therefore  computed 
foi  equidistant  intervals  of  mean  solar  time  at  some  assumed 
meridian,  usually  that  of  the  Greenwich  (Eng.)  Observatory. 

The  Almanac  is  divided  into  three  parts.  The  data  in  Part  I 
are  computed  for  the  meridian  of  Greenwich,  usually  at  the  in- 
stant of  mean  noon,  or  beginning  of  the  Astronomical  day;  this 
section  of  the  almanac  is  arranged  particularly  for  the  conven- 
ience of  navigators.  Part  II  is  calculated  for  the  meridian  of 
the  Washington  Observatory  (longitude  5^  8m  15*. 78  west),  and 

*  Similar  publications  by  other  governments  are:  The  Nautical  Almanac  (Great 
Britain),  Berliner  Astronomisches  Jahrbuch  (Germany),  the  Connaissance  des 
Temps  (France),  and  the  Almanaque  Nautico  (Spain). 

62 


THE   AMERICAN    EPHEMERIS   AND   NAUTICAL  ALMANAC        63 

is  intended  chiefly  for  the  use  of  astronomers  and  geodesists. 
Part  III  contains  data  for  predicting  such  phenomena  as 
eclipses,  occupations,  etc.  At  the  end  of  the  volume  will  be 
found  a  collection  of  tables  which  are  especially  useful  to  the 
surveyor. 

In  addition  to  the  complete  Ephemeris,  there  is  also  published 
a  smaller  volume  which  consists  of  Part  I  and  the  special  tables 
last  mentioned.  A  pamphlet  containing  Azimuths  of  Polaris 
and  a  Solar  Ephemeris  is  published  by  the  General  Land  Office, 
Washington,  D.  C. 

The  data  in  the  first  few  pages  of  Part  I  of  the  Ephemeris  are 
given  for  the  instant  of  mean  noon  at  the  meridian  of  the  Green- 
wich Observatory.  This  instant  of  time  corresponds  to  oh  of  the 
Astronomical  date  given,  or  to  12  M.,  if  it  is  regarded  as  a  Civil 
date.  (See  page  64.)  The  data  include  the  right  ascension, 
declination  and  semidiameter  of  the  sun,  together  with  the  varia- 
tion of  these  quantities  per  hour  of  mean  solar  time.  The 
equation  of  time  and  the  right  ascensions  of  the  mean  sun  are  also 
given.  Following  the  solar  ephemeris  is  the  ephemeris  of  the 
moon,  and  lastly  those  of  the  planets. 

Whenever  the  value  of  a  coordinate  is  to  be  taken  from  the 
Almanac  for  a  given  instant  of  time,  it  is  essential  that  the  Green- 
wich mean  time  be  known  with  sufficient  accuracy  for  the  pur- 
pose. If  the  coordinate  in  question  is  varying  rapidly  the  time 
must  be  known  with  greater  accuracy  than  it  would  if  the  coordi- 
nate were  varying  slowly.  If  the  local  mean  time  is  known,  the 
Greenwich  Mean  Time  is  found  by  adding  the  west  longitude  of 
the  place  expressed  in  hours,  minutes  and  seconds.  If  Standard 
time  is  used,  the  G.  M.  T.  is  found  by  adding  the  longitude  (in 
hours)  of  the  Standard  meridian  to  which  it  refers.  The  tabular 
quantities  are  readily  corrected  for  changes  in  their  values  since 
noon  by  adding  (algebraically)  the  hourly  variation  multiplied 
by  the  number  of  hours  in  the  Greenwich  Mean  Time.  It  should 
be  observed  that  these  quantities  " variation  per  hour"  are  the 
rates  of  change  at  the  instant  of  noon,  i.e.,  they  are  differential 


64 


PRACTICAL  ASTRONOMY 


SUN,  1916 
FOR   GREENWICH   MEAN   NOON 


Date. 

u 

.c 

£.* 

£8 

1* 

Apparent 
Right 
Ascension. 

Var. 
Hour. 

Apparent 
Declina- 
tion. 

Var. 
Hour. 

Semi- 
diam- 
eter. 

Hor. 
Par. 

Equation 
of  Time. 
App.  — 
Mean. 

Var. 
Hour 

Sidereal 
Time,  or 
Right  As- 
cension of 
Mean  Sun. 

Jan.    i 

Sa 

h    m     s 
18  42  27.14 

5 

11.054 

0        1          II 

-23   535  o 

+  11.  21 

16  17.84 

8.95 

m      s 
-  3  10.95 

s 
—  1.198 

h  m     s 
1839  16.19 

2 

Su 

18  46  52.30 

i  i  .  042 

23   052.1 

12.36 

16  17.84 

8.95 

3  39-55 

1.186 

18  43  12  .  75 

3 

Mo 

18  51  17.16 

i  i  .  028 

22554L5 

13.51 

16  17.84 

8.95 

4     7-85 

1.172 

1847    9.3I 

4 

Tu 

18  55  41.67 

11.013 

2250   3.5 

14.65 

16  17.84 

8.95 

4  35-81 

1.  157 

1851    5.86 

5 

We 

19    o    5.81 

10.997 

224358.2 

15.78 

16  17.83 

8.95 

5    3-39 

1.140 

1855    2.42 

6 

Th 

19    429.53 

10.979 

-223725.8 

+16.91 

16  17.82 

8.95 

-  5  30-55 

—  1.  122 

185858.98 

7 

Fr 

19    852.80 

10.960 

22  30  26  .  6 

18.02 

16  17.80 

8-95 

5  57-27 

I.I03 

19    255-54 

8 

Sa 

19  13  15.60 

10.939 

2223    O.? 

19.12 

16  17.78 

8-95 

6  23.50 

1.083 

19    652.10 

9 

Su 

191737.88 

10.917 

2215    8.4 

20.22 

16  17.76 

8.95 

6  49-23 

1.061 

19  1048.66 

10 

Mo 

19  21  59.62 

10.894 

22    649.9 

21.31 

16  17.73 

8.95 

714.41 

1.038 

191445.21 

ii 

Tu 

19  26  2O.80 

10.870 

—2158  5.6 

+22.38 

16  17.70 

8.95 

-  7  39-03 

—  I.OI^ 

191841.77 

12 

We 

193041.39 

10.845 

214855.6 

23-45 

16  17.66 

8.95 

8    3-06 

0.988 

192238.33 

13 

Th 

1935     1-36 

10.819 

21  3920.2 

24.50 

16  17.62 

8.95 

8  26.47 

0.962 

192634.89 

14 

Fr 

193920.70 

10.792 

2I29I9.7 

25.54 

16  17.57 

8.95 

8  49-25 

0.935 

193031.45 

15 

Sa 

194339-39 

10.765 

2II854.5 

26.57 

16  17.52 

8.95 

9  H.38 

0.908 

1934  28.01 

16 

Su 

194757.40 

10.737 

—21   8  4.7 

+27.58 

16  17.46 

8.95 

-  9  32.84 

-0.880 

19  38  24  56 

17 

Mo 

1952  14-73 

10.708 

205650.7 

28.58 

16  17.39 

8.95 

9  53-61 

0.851 

19  42  21.12 

18 

Tu 

I9563L36 

10.678 

204512.8 

29-57 

16  17.32 

8.94 

10  13.69 

0.821 

1946  17.68 

19 

We 

20   o  27  .  28 

10.648 

2033II.3 

30-55 

16  17.24 

8.94 

10  33-05 

0.791 

195014.24 

20 

Th 

20    5    2.47 

10.617 

202046.5 

3I.5I 

16  17.16 

8-94 

10  51.68 

0.761 

195410.79 

21 

Fr 

20    916.92 

10.586 

—  20    758.8 

+32.46 

16  17.07 

8.94 

-ii     9-57 

—0.730 

1958    7.35 

22 

Sa 

20  1330.62 

10.555 

195448.5 

33-40 

16  16.97 

8.94 

ii  26.72 

0.699 

20     2     3.91 

23 

Su 

20  I?  43.57 

10.524 

I94H5.8 

34-32 

16  16.87 

8.94 

ii  43.ii 

0.667 

20    6    0.47 

24 

Mo 

2021  55.76 

10.492 

I9272I.2 

35-23 

16  16.76 

8.94 

ii  58.74 

0.635 

20    957.02 

25 

Tu 

2026     7.18 

10.460 

1913  5-0 

36.12 

16  16.65 

8.94 

12  13.60 

0.603 

201353.58 

26 

We 

2030  17.82 

10.427 

-185827.5 

+37-00 

16  16.53 

8.94 

—  12   27.68 

-0-571 

20  17  50.14 

27 

Th 

2034  27.68 

10.395 

184329.2 

37.86 

16  16.41 

8.94 

12  40.99 

0.538 

20  21  46.69 

28 

Fr 

203836.76 

10.362 

182810.3 

38.71 

16  16.28 

8.94 

12  53-51 

0.505 

20  25  43  25 

29 

Sa 

2042  45.05 

10.329 

181231.2 

39-54 

16  16.15 

8.93 

13     5-24 

0.472 

20  2939.81 

30 

Su 

204652.54 

10.296 

175632.4 

40.35 

16  16.02 

8.93 

13  16.18 

0.439 

20  33  36  .  36 

31 

Mo 

205059.23 

10.262 

—174014.3 

+41.15 

16  15.88 

8.93 

—13  26.31 

—0.406 

203732.92 

Feb.    i 

Tu 

2055    5.  ii 

10.228 

172337.2 

41-93 

16  15.74 

8.93 

13  35.64 

0.372 

20  41  29.48 

2 

We 

2059  10.19 

10.194 

17   641.6 

42.69 

16  15.59 

8.93 

13  44.16 

0.338 

20  45  26.03 

Th 

21    3  14.45 

10.160 

164927.9 

43-44 

i  6  15.44 

8-93 

13  51-86 

0.304 

20  49  22.59 

4 

Fr 

21    7  I7-89 

10.126 

163156.5 

44-17 

16  15.28 

8.03 

13  58.74 

0.270 

205319.15 

s 

Sa 

21  II  20.51 

10.092 

—1614  8.0 

+44-88 

16  15.13 

8.92 

—  14    4.80 

—0.235 

205715.70 

6 

Su 

21  15  22.3O 

10.058 

1556   2.7 

45.56 

16  14.97 

8.92 

14  10.04 

O.2OI 

21      I  12.26 

7 

Mo 

21  1923.27 

10.023 

15374LO 

46.23 

16  14.81 

8.92 

14  14.46 

0.167 

21    5    8.82 

8 

Tu 

21  23  23.42 

9.989 

1519   3-4 

46.89 

16  14.65 

8.92 

14  18.05 

0.133 

21    9    5-37 

9 

We 

21  27  22.75 

9-955 

15    010.3 

47-52 

16  14.48 

8.92 

14  20.83 

0.099 

21  13    1-93 

10 

Th 

21  31  21.28 

9.922 

—  1441     2.2 

+48.14 

16  14.31 

8.92 

—  14  22.8o 

—0.065 

21  16  58.48 

ii 

Fr 

21  35  I9.oo 

I42I39.5 

48.74 

16  14.14 

8.92 

14  23.96 

—0.032 

21  2055.04 

12 

Sa 

21  39  15.92 

9.856 

14     2     26 

49-33 

16  13.96 

8  91 

14  24.33 

+O.OOI 

21  2451-59 

13 

Su 

21  43  12.06 

9.823 

134211  8 

49  89 

16  13.78 

8.91 

14  23.92 

0.033 

21  28  48.15 

14 

Mo 

21  47    7-43 

9-791 

1322    7.7 

50.44 

1  6  13  59 

8.91 

14  22.73 

0.065 

213244.70 

15 

Tu 

21  51     2.04 

9.760 

—  13    150.6 

+50.98 

16  13  40 

8  91 

—  14  20.78 

+0.097 

21  3641.26 

16 

We 

215455.89 

9.729 

—  1241  20.9 

+51.49 

16  13.21 

8.91 

—14  18.08 

+0.128 

21  4037.81 

THE  AMERICAN   EPHEMERIS   AND  NAUTICAL  ALMANAC       65 

coefficients,  not  the  difference  for  i  hour  that  would  be  derived 
from  the  actual  differences  between  the  tabulated  values. 

Part  II  contains  the  apparent  places  of  35  circumpolar  stars 
computed  for  every  day  in  the  year,  and  also  the  position  of  825 
other  stars  (not  near  the  pole)  for  ten-day  intervals  throughout 
the  year.  The  tables  shown  on  pp.  66  and  67  are  taken  from 
the  above-mentioned  star  tables.  The  precession  of  the  equi- 
noxes causes  the  coordinates  of  circumpolars  to  vary  more  rap- 
idly and  more  irregularly  than  those  of  equatorial  stars,  hence  a 
shorter  interval  is  required  in  the  table  in  order  that  interpolated 
coordinates  may  be  obtained  with  sufficient  precision. 

Another  table  in  Part  II  which  is  occasionally  used  by  the  sur- 
veyor is  that  headed  "Moon  Culminations."  This  table  gives 
the  data  needed  for  making  observations  for  longitude  by  the 
lunar  method.  (Art.  88.) 

The  tables  in  the  back  of  the  volume,  already  referred  to, 
include  the  following:  I,  the  correction  to  an  observed  altitude 
of  the  polestar  for  finding  latitude  (see  Art.  69) ;  II,  converting 
sidereal  into  solar  time;  III,  converting  solar  into  sidereal  time; 
IV,  azimuth  of  Polaris  at  Elongation;  VI,  time  intervals  used 
when  observing  on  8  Cassiopeia,  or  f  Urs<z  Majoris,  and  the 
polestar  (see  Art.  99);  VII,  times  of  upper  culmination  (and 
elongations)  of  Polaris. 

39.    Star  Catalogues. 

Whenever  it  becomes  necessary  to  observe  stars  which  are  not 
included  in  the  list  given  in  the  Ephemeris,  their  positions  must 
be  taken  from  one  of  the  star  catalogues.  These  catalogues  give 
the  mean  place  of  each  star  at  some  epoch,  such  as  the  beginning 
of  the  year  1890,  or  1900,  together  with  the  necessary  data  for 
reducing  it  to  the  mean  place  for  any  other  year.  The  mean 
place  of  a  star  is  that  obtained  by  referring  it  to  the  mean  equinox 
at  the  beginning  of  the  year,  that  is,  the  position  it  would  occupy 
if  its  place  were  not  affected  by  the  small  periodic  terms  of  the 
precession. 

The  year  employed  in  such  reductions  is  that  known  as  the 


66 


PRACTICAL  ASTRONOMY 


APPARENT  PLACES  OF  STARS,  1916 
CIRCUMPOLAR  STARS 

FOR  THE  UPPER  TRANSIT  AT   WASHINGTON 


43  H.  Cephei. 

Mag.  4-5 

a  Ursae  Minoris. 
(Polaris.) 
Mag.  2.1 

4  G.  Octantis. 

Mag.  5-6 

Groombridge  750. 
Mag.  6.7 

Groombridge  944. 

Mag.  6.4 

Wash. 

Right 

Decli- 

Wash. 

A   . 

<  « 

Decli- 

Wash. 

A   . 

Decli- 

Wash. 

A   . 

<  G 

Decli- 

Wash. 

A    . 
<  a 

Decli- 

Mean 

As- 

na- 

Mean 

+"« 

na- 

Mean 

+j-2 

na- 

Mean 

*->-2 

na- 

Mean 

.M.2 

na- 

Time. 

cen- 
sion. 

tion. 

Time. 

'r/  ^ 

tion. 

Time. 

f| 

&  ° 

tion. 

Time. 

5s 

tion. 

Time. 

§g 
tf  ° 

tion. 

h   m 

0      / 

h  m 

0     / 

h  m 

c    t 

h  m 

0     / 

h   m 

0       > 

Jan. 

o  56 
5 

+8548 

Jan. 

i  29 

5 

+8851 

Jan. 

i  42 
s 

-8511 

Jan. 

4  9 
j 

+8520 

Jan. 

5  35 

5 

+85  9 

0.3 

60.76 

52.40 

0.3 

5I.6I 

51.67 

0.3 

13-73 

52.16 

0.4 

61.04 

20.08 

o.S 

14.03 

39-07 

1.3 

60.54 

52.50 

1-3 

50.79 

51.80 

i-3 

13-41 

52.20 

1-4 

60.96 

20.34 

i.S 

14-05 

39-35 

2.3 

60.31 

52.60 

2.3 

49.96 

51.95 

2.3 

13.11 

52.20 

2.4 

60.91 

20.62 

2.5 

14-08 

39-63 

3-3 

60.06 

52.72 

3-3 

49-08 

52.11 

3-3 

12.82 

52.17 

3-4 

60.84 

20.93 

3-4 

14.12 

39-94 

4-3 

59-79 

52.85 

4-3 

48.11 

52.29 

4-3 

12.54 

52.12 

4-4 

60.76 

21.24 

4-4 

14.14 

40.27 

S-3 

59-50 

52.97 

5-3 

47.04 

52.45 

5-3 

12.28 

52.07 

5-4 

60.66 

21.57 

5-4 

14  15 

40.61 

6.2 

59.18 

53-00 

6.3 

45-90 

52.60 

6.3 

12.04 

52.01 

6.4 

60.54 

21.89 

6.4 

14.14 

40.98 

7-2 

58.86 

53-11 

7-3 

44-71 

52.71 

7-3 

11.80 

51.98 

7-4 

60.40 

22.19 

7-4 

14.09 

41-33 

8.2 

58.53 

53-13 

8.3 

43-52 

52.80 

8-3 

11-57 

51.98 

8.4 

60.22 

22.46 

8.4 

14-03 

41-67 

9.2 

58.22 

53-13 

9-3 

42.36 

52.87 

9-3 

11-32 

51.97 

94 

60.04 

22.72 

9-4 

13-94 

42.00 

10.2 

57-93 

53-11 

10.3 

41.26 

52.91 

10.3 

11.03 

51.98 

10.4 

59-87 

22.95 

10.4 

13-85 

42.28 

II.  2 

57.64 

53-09 

II.  3 

40.23 

52-93 

11.3 

10.75 

51.98 

ii.  4 

59-69 

23-15 

II.  4 

13-75 

42.56 

12.2 

57-37 

53-09 

12.3 

39-25 

52-95 

12.3 

10.46 

51.97 

12.4 

59-54 

23-34 

12.4 

13.67 

42.80 

13  2 

57-14 

53-oS 

13-3 

38.32 

52.99 

13-3 

10.14 

51.90 

13-4 

59-39 

23.53 

13-4 

13-61 

43.06 

14.2 

56.90 

53.o6 

14.2 

37-40 

53-03 

14-3 

9-83 

51.93 

14.4 

59-25 

23-74 

14.4 

13-55 

43-31 

15-2 

56.65 

53.o6 

15.2 

36.48 

53-07 

15-3 

9-54 

51.88 

15-4 

59-12 

23.96 

15-4 

13.48 

43-57 

16.2 

56.39 

53-07 

16.2 

35-51 

53-13 

16.3 

9-24 

51.80 

16.4 

58.98 

24.18 

16.4 

13-43 

43-85 

17.2 

18.2 

56  12 
55-  &» 

53.08 
53-08 

17.2 
18.2 

34-50 
33-43 

53-19 
53.26 

17.2 
18.2 

8.95 

51-70 
51-59 

17-4 
18.3 

58.83 
58.08 

24.40 
24-65 

17-4 
18.4 

13.38 
13-31 

44-13 
44-43 

19.2 

55-55 

53-08 

19.2 

32.31 

53-31 

19.2 

8^43 

51-49 

19-3 

58.50 

24.90 

19-4 

13  22 

44-75 

20.2 

55.23 

53.o6 

20.  2 

31.14 

53-34 

20.  2 

8.18 

51-37 

20.3 

58.32 

25.16 

20.4 

13-14 

45-07 

21.2 

54-91 

53  02 

21.2 

29.94 

53-35 

21.2 

7-93 

51.26 

21.3 

58.10 

25-39 

21.4 

I3.O2 

45-39 

22.2 

54-59 

52.96 

22.2 

28.74 

53-35 

22.2 

7.69 

51.16 

22.3 

67-88 

25.60 

22.4 

12.89 

45.68 

23.2 

54-28 

52.88 

23.2 

27-55 

53-32 

23.2 

7-44 

51.07 

23-3 

57.65 

25-79 

23-4 

12.74 

45.96 

24.2 

53.98 

52.78 

24.2 

26.40 

53-28 

24.2 

7.18 

51-00 

24-3 

57-42 

25.98 

24-4 

12.57 

46.22 

25.2 

53-70 

52.65 

25.2 

25  32 

53-21 

25.2 

6.91 

50.92 

25  3 

57-19 

26.13 

25.4 

12.43 

46.46 

26.2 

53  45 

52.54 

26.2 

24  32 

53-15 

26.2 

6.63 

50.84 

26.3 

56.98 

26.26 

26.4 

12.27 

46.67 

27.2 

53-21 

52.43 

27.2 

23.39 

53-09 

27.2 

6.32 

50.75 

27-3 

56.78 

26.38 

27-4 

12.13 

46.87 

28.2 

5298 

52-35 

28.2 

22.51 

53-05 

28.2 

6  01 

50.62 

28.3 

56.59 

26.40 

28.4 

12.00 

47.08 

29.2 

52.76 

52.27 

29.2 

21   63 

53-01 

29.2 

5-70 

50.47 

29.3 

56.43 

26.63 

29.4 

11.90 

47-30 

30  2 

52  54 

52.19 

30.2 

20.73 

53-00 

30.2 

5.4i 

50.29 

30.3 

56.27 

26.80 

30.4 

II.8I 

47.52 

31-2 

52.30 

52   14 

31-2 

19.77 

52.99 

31-2 

5-14 

50.09 

31-3 

56.09 

26.97 

31  4 

11.71 

47-77 

13.70       +13.67 

50.47  +50.46 

11.95  -11.90 

12.31   +12.27 

11.86   +11.  81 

o      57m     i".  657 
+85°    48'    25".  87 

I*  29'"  44"  254 
+88°  51'  25".  03 

ift    42™    6M02 
-85°    ii'  39"  58 

4*     9'«  44"  952 
+85°   20'    i".04 

5n   34m  54«.oi4 
+85°     9'  28".  07    i 

THE  AMERICAN  EPHEMERIS  AND   NAUTICAL  ALMANAC    67 


APPARENT  PLACES  OF  STARS,  1916 

FOR  THE  UPPER  TRANSIT  AT  WASHINGTON 


68  PRACTICAL  ASTRONOMY 

Besselian  fictitious  year.  It  begins  when  the  sun's  mean  longi- 
tude (arc  of  the  ecliptic)  is  280°,  that  is  when  the  right  ascension 
of  the  mean  sun  is  i8A  40™,  which  occurs  about  January  i.  After 
the  catalogued  position  of  the  star  has  been  brought  up  to  the 
mean  place  at  the  beginning  of  the  given  year,  it  must  still  be 
reduced  to  its  "  apparent  place,"  for  the  exact  date  of  the  obser- 
vation, by  employing  formulae  and  tables  given  for  the  purpose 
in  Part  II  of  the  Ephemeris. 

There  are  many  star  catalogues,  some  containing  the  posi- 
tions of  a  very  large  number  of  stars,  but  determined  with  rather 
inferior  accuracy;  others  contain  a  relatively  small  number  of 
stars,  but  whose  places  are  determined  with  the  greatest  accuracy. 
Among  the  best  of  these  latter  may  be  mentioned  the  Greenwich 
ten-year  (and  other)  catalogues,  and  Boss'  Catalogue  of  6188 
stars  for  the  epoch  1900.  (Washington,  1910.) 

For  time  and  longitude  observations,  the  list  given  in  the 
Ephemeris  is  sufficient,  but  for  special  kinds  of  work  where  the 
observer  has  but  a  limited  choice  of  positions,  such  as  finding 
latitude  by  Talcott's  method,  many  other  stars  must  be  ob- 
served. 

40.    Interpolation. 

When  taking  data  from  the  Ephemeris  corresponding  to  any 
given  instant  of  Greenwich  Mean  Time,  it  will  generally  be  neces- 
sary to  interpolate  between  the  tabulated  values  of  the  function. 
The  method  employed  for  performing  this  interpolation  will  de- 
pend upon  the  rapidity  of  the  changes  in  the  function  and  upon 
the  precision  demanded.  The  simplest  method  of  interpolating 
consists  in  assuming  that  the  function  varies  uniformly  between 
two  successive  tabular  values,  and  giving  the  tabulated  function 
an  increase  directly  proportional  to  the  time  elapsed  since  Green- 
wich Noon.  If  the  function  is  represented  graphically,  it  will  be 
seen  that  this  process  places  the  computed  value  on  a  chord  of 
the  function  curve. 

Since,  however,  the  "variation  per  hour"  or  differential  co- 
efficient is  usually  given,  it  is  simpler  to  employ  this  quantity  as 


THE  AMERICAN  EPHEMERIS   AND   NAUTICAL  ALMANAC     69 


the  rate  of  change  of  the  function  and  to  multiply  it  by  the  time 
elapsed  since  noon.  An  examination  of  the  diagram  (Fig.  38a) 
will  show  that  this  is  also  the  more  accurate  method  of  the 
two,  provided  we  always  work  from  the  nearer  tabulated  value, 
because  the  differential  coefficient  gives  the  slope  of  the  tangent 
line,  and  the  curve  lies  nearer  to  the  tangent  line  than  it  does 
to  the  chord. 


Greenwich  Mean  Time 
FIG.  38a. 


Feb.  2 


To  illustrate  these  methods  of  interpolating,  let  it  be  assumed 
that  it  is  required  to  compute  the  sun's  declination  at  22h  G.  M.  T. 
Feb.  i,  the  tabulated  values  for  (noon)  Feb.  i  and  Feb.  2  being  as 
follows: 


Feb. 

i 

2 


Sun's  declination          Variation  per  i* 

S  17°  24/o4//.o  +41". 88 

17    07  09   .8  +  42   .64 

First,  if  we  interpolate  between  the  tabular  values,  without 
reference  to  the  variation  per  hour,  we  obtain  S  17°  08'  34".3,  a 


70  PRACTICAL  ASTRONOMY 

point  on  the  chord,  as  shown  in  Fig.  38a.  Next,  interpolating 
backward  from  the  noon  of  Feb.  2,  using  for  the  variation  per 
hour  the  value  42". 64  and  multiplying  by  2A,  we  find  for  the  dec- 
lination 817°  08'  35".!,  which  is  represented  by  a  point  on  that 
tangent  to  the  curve  which  corresponds  to  Feb.  2.  Working 
from  noon  of  Feb.  i,  using  4i".88,  we  obtain  S  17°  08'  42". 6 
which  gives  a  point  on  the  other  tangent  line;  this  latter  is 
obviously  much  more  in  error  than  either  of  the  other  two 
values. 

Whenever  the  above  processes  fail  to  give  sufficient  precision, 
it  becomes  necessary  to  consider  the  variations  in  the  "  variation 
per  hour."  If  we  imagine  a  parabola  with  its  axis  vertical  and 
so  placed  that  it  passes  through  the  two  given  points  of  the  curve, 
then  it  may  be  shown  that  the  following  process  of  interpolation 
gives  a  point  on  the  parabola  itself  and  consequently  a  result 
which  is  very  close  to  the  true  value  of  the  function.  Since  the 
second  differential  coefficient  of  the  parabola  is  constant,  we  may 
obtain  the  slope  of  the  parabola  at  any  desired  point  by  simple 
interpolation  between  the  given  values  of  the  first  differential 
coefficients.  If  we  obtain  the  value  of  dy/dx  for  a  point  whose 
abscissa  is  half  way  between  the  tabulated  time  and  the  time  for 
which  the  function  is  desired,  there  results  the  slope  of  a  chord 
of  the  parabola  which  extends  from  the  point  representing  the 
tabulated  quantity  to  the  point  representing  the  value  desired, 
because  such  a  chord  may  be  proved  to  be  exactly  parallel  to  the 
tangent  (slope)  so  found.  Hence,  by  finding  the  value  of  the 
"variation  per  hour"  corresponding  to  the  middle  of  the  time 
interval  over  which  we  are  interpolating  and  employing  this  in 
place  of  the  given  "variation  per  hour"  we  place  our  point  on 
the  parabola,  which  closely  coincides  with  the  function  curve 
between  the  adjacent  tabulated  values.  In  the  preceding  ex- 
ample, this  method  gives,  for  the  interpolated  "variation  per 
hour,"  +  42".6i,  and  for  the  declination  S  17°  08'  35".o 
(Fig.38a). 

As  another  example  of  the  process,  let  it  be  required  to  find 


THE  AMERICAN   EPHEMERIS  AND   NAUTICAL  ALMANAC     71 

the  right  ascension  of  the  moon  at  2ih  40™.     The  Ephemeris  of 
the  moon  gives  the  following  data : 

Gr.  M.  T.  R.  A.  of  Moon        Variation  per  min. 

2iA  21*  oom  i7s.55  2a.oi38 

22  21  O2   l8  .20  2  ,OO8O 

23  21  04   l8  .51  2  .OO23 

The  G.  M.  T.  at  the  middle  of  the  interval  between  21*  and 
2ih  40™  is  2ih  2om,  one-third  the  way  from  the  first  to  the  second 
tabular  values.  The  interpolated  "variation  per  min."  for  this 
instant  is,  therefore,  2^.0119.  The  correction  to  the  R.  A.  at 
21*  is  2". 01  19  X  40™  =  80.476,  or  im  20*48.  The  R.  A.  at 
2ih  4om  is,  therefore,  2ih  oim  388.O3. 

For  more  general  interpolation  formulae  the  student  is  referred 
to  Chauvenet's  Spherical  and  Practical  Astronomy,  Doolittle's 
Practical  Astronomy,  Hayford's  Geodetic  Astronomy  and  Rice's 
Theory  and  Practice  of  Interpolation. 

Questions  and  Problems 

1.  Compute  the  sun's  apparent  declination  when  the  Mean  Local  Time  is  8h  30** 
A.M.,  Jan.  16,  1916,  at  a  place  85°  west  of  Greenwich  (see  p.  64). 

2.  Compute  the  right  ascension  of  the  mean  sun  at  local  mean  noon  Jan.  10, 
1916,  at  a  place  96°  10'  west  of  Greenwich. 

3.  Compute  the  equation  of  time  for  local  apparent  noon  Jan.  30,  1916,  at  a 
place  20°  east  of  Greenwich. 

4.  What  is  the  relation  between  the  "right  ascension  of  the  mean  sun"  and  the 
"apparent  right  ascension"  of  the  sun  at  G.  M.  N.  on  Jan.  i,  1916? 

5.  Compute  the  right  ascension  of  the  sun  at  G.  M.  T.  10*  by  the  four  different 
methods  explained  in  Art.  40. 


CHAPTER  VII 

THE  EARTH'S  FIGURE  — CORRECTIONS  TO   OBSERVED 

ALTITUDES 

41.  The  Earth's  Figure. 

The  earth's  form  is  approximately  that  of  an  oblate  spheroid 
whose  shortest  axis  is  the  axis  of  rotation.  The  actual  figure 
deviates  slightly  from  that  of  a  perfect  spheroid,  but  for  most 
astronomical  purposes  these  deviations  may  be  disregarded. 
Each  meridian  may  therefore  be  considered  as  an  ellipse,  and 
the  equator  and  all  parallels  of  latitude  as  circles.  The  semi- 
major  axis  of  the  meridian  ellipse  is  about  3962.80  miles,  and  the 
semi-minor  axis  is  3949.56  miles  in  length.  The  length  of  i°  of 
latitude  at  the  equator  is  68.704  miles;  at  the  pole  it  is  69.407 
miles. 

In  locating  points  on  the  earth's  surface  by  means  of  coordi- 
nates there  are  three  kinds  of  latitude  to  be  considered.  The 
latitude  as  found  by  astronomical  observation  is  dependent  upon 
the  direction  of  gravity  as  indicated  by  the  spirit  levels  of  the 
instrument,  and  is  affected  by  any  abnormal  deviations  of  the 
plumb  line*  at  this  point;  the  latitude  as  found  directly  by  ob- 
servations is  called  the  astronomical  latitude.  The  geodetic 
latitude  is  the  latitude  that  would  be  found  by  observation  if 
the  plumb  line  were  normal  to  the  surface  of  the  spheroid  taken 
to  represent  the  earth's  figure,  that  is,  if  all  of  the  irregularities 
of  the  surface  were  smoothed  out.  Evidently  the  geodetic 
latitude  cannot  be  directly  observed  but  must  be  found  by  com- 
putation. The  geocentric  latitude  is  the  angle  between  the  plane 
of  the  equator  and  a  line  drawn  from  the  centre  of  the  earth  to 
the  point  on  the  surface.  In  Fig.  39  the  line  AD  is  normal  to 

*  These  deviations  are  small,  averaging  about  3"  or  4",  but  in  some  cases 
deviations  of  nearly  30"  are  found. 

72 


THE  EARTH'S  FIGURE 


73 


the  earth's  surface  at  A,  and  the  angle  ABE  is  the  geodetic 
latitude  of  A.  If  the  plumb  line  coincides  with  AD,  this  is 
also  the  astronomical  latitude.  The  angle  ACE  is  the  geocen- 
tric latitude.  The  difference  between  the  two,  or  angle  BAC, 


is  called  the  angle  of  the  vertical,  or  the  reduction  of  latitude 
The  geocentric  latitude  is  always  less  than  the  observed  lati- 
tude by  an  angle  which  varies  from  about  o°  n'  30"  in  latitude 
45°  to  zero  at  the  equator  and  the  poles.  Whenever  observations 
are  made  at  any  point  on  the  earth's  surface  it  is  necessary  to 
reduce  the  observed  values  to  their  values  at  the  earth's  centre 
before  they  can  be  combined  with  other  data  referred  to  the 
centre.  In  making  this  reduction  the  geocentric  latitude  must 
be  used  if  the  exact  position  of  the  observer  with  reference  to  the 
centre  is  to  be  computed.  For  most  of  the  observations  treated 
in  the  following  chapters  it  will  not  be  necessary  to  consider  the 
spheroidal  shape  of  the  earth;  it  will  be  sufficiently  exact  to  regard 
it  as  a  sphere. 

42.  Parallax. 

The  coordinates  of  a  celestial  object  as  given  in  the  Ephemeris 
are  referred  to  the  centre  of  the  earth,  while  the  coordinates 


74 


PRACTICAL  ASTRONOMY 


obtained  by  observation  are  necessarily  measured  from  a  point 
on  the  surface,  and  must  be  reduced  to  the  centre.  The  case 
most  frequently  occurring  in  practice  is  that  in  which  the  altitude 
of  an  object  is  observed  and  the  geocentric  altitude  is  desired. 
For  all  objects  except  the  moon  the  distance  of  the  body  is  so 
great  that  it  is  sufficiently  accurate  to  regard  the  earth  as  a 


FIG.  40 

sphere.  In  Fig.  40,  the  angle  ZOS  is  the  observed  zenith  dis- 
tance, or  the  complement  of  the  observed  altitude,  and  ZCS  is 
the  true  zenith  distance.  This  apparent  displacement  of  the 
object  on  the  celestial  sphere  is  called  parallax.  The  effect  of 
parallax  is  simply  to  decrease  the  altitude  without  altering  the 
azimuth  of  the  body,  provided  the  spheroidal  form  of  the  earth 
be  disregarded.  The  difference  in  direction  between  the  lines 
OS  and  CS,  or  the  angle  OSC,  is  the  parallax  correction.  In  the 
triangle  OSC,  angle  COS  may  be  considered  as  known,  since  the 
altitude  or  complement  of  ZOS  is  observed.  The  distance  OC 
is  the  semidiameter  of  the  earth  (3958  miles),  and  CS  is  the 


THE   EARTH'S   FIGURE  75 

distance  from  the  earth's  centre  to  the  centre  of  the  body  ob- 
served. Solving  this  triangle, 

OC* 

sin  S  =  sin  ZOS  X  TT^T  •  [50] 

It  is  evident  that  the  parallax  correction  will  be  zero  at  the 
zenith  and  a  maximum  at  the  horizon.  For  the  maximum 
value  of  the  correction,  that  is,  when  ZOS  =  90°,  we  have 

OC 
Sm      =  GS~' 

which  is  a  constant  for  all  places  if  the  earth  is  considered  spher- 
ical. In  order  to  find  this  correction  for  the  altitude  of  S  it  is 
convenient  to  compute  it  from  the  maximum  value,  or  parallax 

OC 

on  the  horizon.     If  we  put  Ph  for  — -  (the  horizontal  parallax) 

Coi 

then  equation  [50]  may  be  written 

sin  5  =  sin  Ph  sin  z  =  sin  Ph  cos  h,  [52] 

where  h  is  the  apparent  altitude  of  the  object.  But  S  and  Ph 
are  usually  very  small  angles,  and  the  error  is  negligible  if  the 
sines  are  replaced  by  their  arcs.*  Equation  [52]  then  becomes 

S"  =  Pk"  cos  h,  [53] 

where  S"  and  Ph"  are  both  in  seconds  of  arc. 

For  the  moon  the  mean  value  of  the  horizontal  parallax  is 
about  o°  57'  02"  ;    for  the  sun  it  is  8".8;  for  the  fixed  stars  it  is 

*  The  sine  may  be  expressed  as  a  series  as  follows: 

x3       x5 
smx  =  x  —  T~  +  T~  —  •  •  '  [54! 

U       UL 

Replacing  sin  x  by  *  amounts  to  neglecting  all  terms  after  the  first.  Whether  the 
error  will  be  appreciable  in  any  given  case  may  be  determined  by  computing  the 
value  of  the  first  of  the  neglected  terms.  If  x  =  i°  the  neglected  terms  are  less 
than  .005  of  i%  of  x.  The  error  in  an  angle  of  i°  would  be  less  than  o".2.  The 
moon  is  the  only  object  whose  parallax  is  nearly  as  large  as  i°,  so  that  for  all  other 
objects  this  approximation  is  usually  allowable.  Similarly  for  cos  x  =  i,  the  terms 
neglected  are  those  of  the  series 

x*        x* 
COS  x  =  I  -  r-~  +  1 '  '  *  (SSI 

UL    11 


7°  PRACTICAL  ASTRONOMY 

too  small  to  be  detected.  The  horizontal  parallaxes  of  objects 
in  the  solar  system  are  given  in  the  Nautical  Almanac.*  For  the 
parallax  of  the  sun  for  different  altitudes  see  Table  IV  (A). 

43.   Refraction. 

Refraction  is  the  term  applied  to  the  bending  of  a  ray  of  light 
by  the  atmosphere  as  it  passes  from  a  celestial  object  to  the 
observer's  eye.  On  account  of  the  increasing  density  of  the 
layers  of  air  the  rays  of  light  coming  from  any  object  are  bent 
downward  into  a  curve,  and  consequently  when  the  rays  enter 
the  eye  they  have  a  greater  inclination  to  the  horizon  than  they 
did  before  entering  the  atmosphere.  For  this  reason  all  objects 
appear  higher  above  the  horizon  than  they  actually  are.  In 


FIG.  41 

Fig.  41,  S  is  the  true  position  of  a  star  and  S'  its  apparent  position. 
The  light  from  5  is  bent  into  a  curve  aO,  and  the  star  is  seen  in 
the  direction  of  the  tangent  ObS'.  The  angle  which  must  be 
subtracted  from  the  altitude  of  5'  to  obtain  the  altitude  of  S  is 
called  the  refraction  correction.  This  angle  is  really  the  angle 
SOS',  but  on  account  of  the  great  distance  of  celestial  objects 

*  On  account  of  the  spheroidal  form  of  the  earth  the  equatorial  diameter  is  the 
greatest  and  the  parallax  at  the  equator  is  a  maximum;  the  parallaxes  are  therefore 
given  in  the  Ephemeris  under  the  heading  "  Equatorial  Horizontal  Parallax." 


THE  EARTH'S  FIGURE  77 

and  the  small  angle  of  refraction  the  correction  may  be  con- 
sidered as  the  angle  SbS'.  From  the  figure  it  is  evident  that 

ZcS  =  ZOS'  +  S'bS, 

or  zr  =  z  +  r,  [56] 

where  zf  =  the  true  and  z  =  the  apparent  zenith  distance  and 
r  =  the  refraction  correction.  The  approximate  law  of  astro- 
nomical refraction  may  be  deduced  by  assuming  that  the  bend- 
ing all  occurs  at  point  b.  The  general  law  of  refraction,  when  a 
ray  enters  a  refracting  medium,  is  expressed  by  the  equation 

sin  zf  —  n  sin  z,  [57] 

where  n  is  the  index  of  refraction  of  the  given  medium;  for  air 
its  value  is  roughly  about  1.00029. 

Substituting  from  Equa.  [56], 

sin  (z  +  r)  =  n  sin  z,  [58] 

Expanding,         sin  z  cos  r  +  cos  z  sin  r  =  n  sin  z.  [59] 

Since  r  is  a  small  angle  (never  greater  than  40')  it  is  allowable 
to  put  cos  r  =  i  and  sin  r  =  r\  then 

sin  z  +  r  cos  z  =  n  sin  z, 

and  r  cos  z  =  (n  —  i)  sin  z, 

or  r  =  (n  —  i)  tan  z.  [60] 

Replacing  n  by  1.00029  and  dividing  by  arc  i"  to  reduce  r  from 
circular  measure  to  seconds  of  arc, 

r"  =  .  ('00°29) ,  tan  « 

(.000,0048) 

=  60"  tan  z 

=  60"  cot/*,  [61] 

where  h  is  the  apparent  altitude.* 

The  value  of  n  varies  considerably  with  the  temperature  and 
the  pressure  of  the  air,  so  that  equation  [61]  must  be  considered 
as  giving  only  a  rough  approximation  to  the  true  refraction. 

*  "  Apparent "  is  used  here  simply  to  distinguish  between  the  direction  of  the  star 
as  actually  seen  and  the  direction  unaffected  by  refraction.  In  speaking  of  parallax, 
the  word  "  apparent  "  has  a  different  meaning,  and  in  case  of  aberration,  still 
another  meaning. 


7  8  PRACTICAL  ASTRONOMY 

For  high  altitudes  this  formula  is  nearly  correct,  but  for  altitudes 
under  10°  it  is  not  sufficiently  exact.  If  both  sides  of  the  equa- 
tion are  divided  by  60  so  that  r  is  reduced  to  minutes,  we  have 
the  extremely  simple  relation  that  the  refraction  in  minutes  equals 
the  natural  cotangent  of  the  altitude.  For  altitudes  measured 
with  an  engineer's  transit  this  formula  is  close  enough  for  alti- 
tudes greater  than  about  10°.  For  more  accurate  values  of  the 
refraction  Table  I  may  be  used.  From  the  table  it  will  be  seen 
that  the  refraction  correction  is  zero  at  the  zenith,  about  i'  at 
an  altitude  of  45°,  and  about  o°  34'  at  the  horizon.* 

The  following  formula,  due  to  Professor  George  C.  Corns tock, 
gives  very  accurate  values  of  the  refraction  for  altitudes  greater 
than  20°,  and  is  sufficiently  accurate  for  all  field  observations 
made  with  surveyors'  instruments. 

r"  =  -2&A-cotA,  [62] 

460  +  / 

in  which  b  is  the  barometer  reading  in  inches,  and  /  is  the  tem- 
perature in  Fahrenheit  degrees. 
Example. 

Altitude  30°,  barometer  29.1™',  thermometer  81°  F. 
log.  983  =  2.9926 
log.  29.1  =  1.4639 

460°  colog.  541  =  7.2668 

8 1  cot  h  =  0.2386 

541°  1.9619 

r  =      9i".6 
=  i'3i".6 

44.   Semidiameters. 

The  discs  of  the  sun  and  moon  are  circular,  and  their  angular 
semidiameters  are  given  for  each  day  in  the  Ephemeris.  Since 
measurements  can  only  be  taken  to  the  edge,  or  limb,  the  altitude 
of  the  centre  of  the  object  is  obtained  by  making  a  correction 

*  The  sun's  diameter  is  about  32',  slightly  less  than  the  refraction  on  the  horizon; 
when  the  sun  has  actually  gone  below  the  horizon  at  sunset  the  entire  disc  is  still 
visible  on  account  of  the  34'  increase  in  its  apparent  altitude  due  to  atmospheric 
refraction. 


THE   EARTH'S    FIGURE 


79 


equal  to  the  semidiameter.  The  apparent  angular  semidiameters 
given  in  the  Ephemeris  may  be  affected  in  two  ways,  one  by  the 
change  in  the  observer's  distance  because  he  is  on  the  earth's 
surface,  the  other  by  the  difference  in  the  amount  of  refraction 
correction  on  the  upper  and  lower  edges  of  the  disc. 

The  semidiameter  given  in  the  Ephemeris  is  that  as  seen  from 
the  centre  of  the  earth.  When  the  object  is  in  the  zenith  the 
observer  is  nearly  4000  miles  nearer  than  when  it  is  in  the  hori- 
zon. The  moon  is  about  240,000  miles  distant  from  the  earth, 
so  that  the  semidiameter  is  increased  by  about  ^V  part,  or 
about  16". 

The  vertical  diameter  of  an  object  appears  to  be  less  than  its 
horizontal  diameter  because  the  refraction  lifts  the  lower  edge 
more  than  it  does  the  upper  edge.  The  disc  then  presents  the 
appearance  of  an  ellipse.  When  the  sun  is  rising  or  setting,  the 
contraction  is  most  noticeable.  This  contraction  of  the  semi- 
diameter  does  not  affect  the  correction  to  an  observed  altitude, 
but  must  be  taken  into  account  when  the  distance  is  measured 
between  the  moon's  limb  and  a  star  or  a  planet.  (See  Art.  108.) 

For  the  angular  semidiameter  of 
the  sun  on  the  first  day  of  each 
month  see  Table  IV  (B). 

45-   Dip- 

If  altitudes  are  taken  from  the 
sea  horizon,  as  when  observing 
on  board  ship  with  the  sextant, 
the  measured  altitude  must  be 
diminished  by  the  angular  dip 
of  the  sea  horizon  below  the  true 
horizon.  In  Fig.  42  suppose  the 
observer  to  be  at  O;  the  true 
horizon  is  OB  and  the  sea  horizon  FIG 

OH.    Let  OP  =  h,  the  height  in 

feet  above  the  surface;  PC  =  R,  the  radius  of  the  earth;  and 
D,  the  angle  of  dip. 


— B 


8O  PRACTICAL  ASTRONOMY 


Then  cosZ>  =  --^-  [63] 

jy. 
Putting  cos  D  =  i  —  —  ,  neglecting  other  terms  in  the  series, 

D2         h  h 


Replacing  R  by  its  value  in  feet,  20,884,000,  and  dividing  by 
sin  i  '.to  reduce  D  to  minutes, 


/  - 

V  -  X  sin  i' 

"     2 

=  i.o64Vh.  [64] 

This  shows  the  amount  of  dip  unaffected  by  refraction.  The 
effect  of  refraction  is  to  apparently  lift  the  horizon,  and  the  dip 
affecting  the  observed  altitude  is  therefore  less  than  that  given 
by  the  formula.  If  the  coefficient  1.064  is  taken  as  unity,  the 
formula  is  nearer  the  truth  and  is  simpler,  although  still  some- 
what too  large.  Table  IV  (C)  ,  based  on  a  more  exact  formula,  will 
be  seen  to  give  smaller  values.  For  ordinary  sextant  observations 
made  at  sea,  where  the  greatest  precision  is  not  required,  it  is 
sufficient  to  take  the  dip  in  minutes  equal  to  the  square  root  of 
the  height  of  the  eye  in  feet,  that  is, 

D'  =  V/Tft.  [65] 

46.   Sequence  of  Corrections. 

Strictly  speaking,  the  corrections  to  the  altitude  should  be 
made  in  the  following  order  : 

(i)  Instrumental  corrections;  (2)  dip  (if  at  sea);  (3)  refraction; 
(4)  semidiameter;  (5)  parallax.  In  practice,  however,  it  is  not 
always  necessary  to  follow  this  order  exactly.  At  sea  the  cor- 
rections are  often  taken  together  as  a  single  "  correction  to  the 
altitude."  Care  should  be  taken  to  use  the  refraction  correction 


THE  EARTH'S   FIGURE  8 1 

for  the  limb  observed,  not  for  the  centre,  for  if  the  altitude  is 
small  the  two  will  differ  appreciably. 

Problems 

1.  Compute  the  sun's  mean  horizontal  parallax.    The  sun's  mean  distance  is 
92,900,000  miles;  for  the  earth's  radius  see  Art.  41.     Compute  the  sun's  parallax 
at  an  altitude  of  60°. 

2.  Compute  the  moon's  mean  horizontal  parallax.     The  moon's  mean  distance 
is  238  800  miles;  for  the  earth's  radius  see  Art.  41.     Compute  the  moon's  parallax 
at  an  altitude  of  45°, 

3.  Estimate  the  error  in  formula  [53]  due  to  placing  the  arc  equal  to  the  sine. 

4.  Estimate  the  error  in  formula  [61]  due  to  placing  cos  r  =  i  and  sin  r  =  r. 

5.  Compute  the  refraction  correction  for  an  altitude  of  20°,  when  the  temperature 
is  60°  F.  and  the  barometer  reads  28.6  inches. 


CHAPTER  VIII 
DESCRIPTION   OF   INSTRUMENTS 

47.   The  Engineer's  Transit. 

The  engineer's  transit  is  an  instrument  for  measuring  hori- 
zontal and  vertical  angles.  For  the  purpose  of  discussing  the 
theory  of  the  instrument  it  may  be  regarded  as  a  telescopic  line 
of  sight  having  motion  about  two  axes  at  right  angles  to  each 
other,  one  vertical,  the  other  horizontal.  The  line  of  sight  is 
determined  by  the  optical  centre  of  the  object  glass  and  the 
intersection  of  two  cross  hairs*  placed  in  its  principal  focus. 
The  vertical  axis  of  the  instrument  coincides  with  the  axes  of 
two  spindles,  one  inside  the  other,  each  of  which  is  attached  to  a 
horizontal  circular  plate.  The  lower  plate  carries  a  graduated 
circle  for  measuring  horizontal  angles;  the  upper  plate  has  two 
verniers,  on  opposite  sides,  for  reading  angles  on  the  circle. 
On  the  top  of  the  upper  plate  are  two  uprights,  or  standards, 
supporting  the  horizontal  axis  to  which  the  telescope  is  attached 
and  about  which  it  rotates.  At  one  end  of  the  horizontal  axis 
is  a  vertical  arc,  or  a  circle,  and  on  the  standard  is  a  vernier,  in 
contact  with  the  circle,  for  reading  the  angles.  The  plates  and 
the  horizontal  axis  are  provided  with  clamps  and  slow-motion 
screws  to  control  the  motion.  On  the  upper  plate  are  two  spirit 
levels  for  levelling  the  instrument,  or,  in  other  words,  for  making 
the  vertical  axis  coincide  with  the  direction  of  gravity. 

The  whole  instrument  may  be  made  to  turn  in  a  horizontal 
plane  by  a  motion  about  the  vertical  axis,  and  the  telescope  may 
be  made  to  move  in  a  vertical  plane  by  a  motion  about  the 
horizontal  axis.  By  means  of  a  combination  of  these  two 

*  Also  called  wires  or  threads;  they  are  either  made  of  spider  threads  or  are 
lines  ruled  upon  glass. 

82 


DESCRIPTION  OF  INSTRUMENTS  83 

motions,  vertical  and  horizontal,  the  line  of  sight  may  be  made 
to  point  in  any  desired  direction.  The  motion  of  the  line  of 
sight  in  a  horizontal  plane  is  measured  by  the  angle  passed  over 
by  the  index  of  the  vernier  along  the  graduated  horizontal 
circle.  The  angular  motion  in  a  vertical  plane  is  measured  by 
the  angle  on  the  vertical  arc  indicated  by  the  vernier  attached 
to  the  standard.  The  direction  of  the  horizon  is  denned  by 
means  of  a  long  spirit  level  attached  to  the  telescope.  When 
the  bubble  is  central  the  line  of  sight  should  lie  in  the  plane  of 
the  horizon.  To  be  in  perfect  adjustment,  (i)  the  axis  of  each 
spirit  level  *  should  be  in  a  plane  at  right  angles  to  the  vertical 
axis;  (2)  the  horizontal  axis  should  be  at  right  angles  to  the 
vertical  axis;  (3)  the  line  of  sight  should  be  at  right  angles  to  the 
horizontal  axis;  (4)  the  axis  of  the  telescope  level  should  be 
parallel  to  the  line  of  sight,  and  (5)  the  vernier  of  the  vertical 
arc  should  read  zero  when  the  bubble  is  in  the  centre  of  the  level 
tube  attached  to  the  telescope.  When  the  plate  levels  are 
brought  to  the  centres  of  their  tubes,  and  the  lower  plate  is  so 
turned  that  the  vernier  reads  o°  when  the  telescope  points  south, 
then  the  vernier  readings  of  the  horizontal  plate  and  the  vertical 
arc  for  any  position  of  the  telescope  are  coordinates  of  the 
horizon  system  (Art.  12).  If  the  horizontal  circles  are  clamped 
in  any  position  and  the  telescope  is  moved  through  a  complete 
revolution,  the  line  of  sight  describes  a  vertical  circle  on  the 
celestial  sphere.  If  the  telescope  is  clamped  at  any  altitude  and 
the  instrument  turned  about  the  vertical  axis,  the  line  of  sight 
describes  a  cone  and  traces  out  on  the  sphere  a  circle  of  equal 
altitudes,  or  an  almucantar. 

48.   Elimination  of  Errors. 

It  is  usually  more  difficult  to  measure  an  altitude  accurately 
with  the  transit  than  to  measure  a  horizontal  angle.  While  the 
precision  of  horizontal  angles  may  be  increased  by  means  of 
repetitions,  in  measuring  altitudes  the  precision  cannot  be 

*  The  axis  of  a  level  may  be  defined  as  a  line  tangent  to  the  curve  of  the  glass 
tube  at  the  point  on  the  scale  taken  as  the  zero  point,  or  at  the  centre  of  the  tube. 


84  PRACTICAL  ASTRONOMY 

increased  by  repeating  the  angles,  owing  to  the  construction  of 
the  instrument.     The  vertical  arc  usually  has  but  one  vernier, 
so  that  the  eccentricity  cannot  be  eliminated,  and  this  vernier 
often  does  not  read  as  closely  as  the  horizontal  vernier.     One 
of  the  errors,  which  is  likely  to  be  large,  but  which  may  be  elimi- 
nated readily,  is  that  known  as  the  index  error.     The  measured 
altitude  of  an  object  may  differ  from  the  true  reading  for  two 
reasons:  first,  the  zero  of  the  vernier  may  not  coincide  with  the 
zero  of  the  circle  when  the  telescope  bubble  is  in  the  centre  of 
its  tube;  second,  the  line  of  sight  may  not  be  horizontal  when 
the  bubble  is  in  the  centre  of  the  tube.     The  first  part  of  this 
error  can  be  corrected  by  simply  noting  the  vernier  reading  when 
the  bubble  is  central,  and  applying  this  as  a  correction  to  the 
measured  altitude.     To  eliminate  the  second  part  of  the  error 
the  altitude  may  be  measured  twice,  once  from  the  point  on  the 
horizon  directly  beneath  the  object  observed,  and  again  from 
the  opposite  point  of  the  horizon.     In  other  words,  the  instru- 
ment may  be  reversed  (180°)  about  its  vertical  axis  and  the 
vertical  circle  read  in  each  position  while  the  horizontal  cross 
hair  of  the  telescope  is  sighting  the  object.     The  mean  of  the 
two  readings  is  free  from  the  error  in  the  sight  line.     Evidently 
this  method  is  practicable  only  with  an  instrument  having  a 
complete  vertical  circle.     If  the  reversal  is  made  in  this  manner 
the  error  due  to  non-adjustment  of  the  vernier  is  eliminated  at 
the  same  time,  so  that  it  is  unnecessary  to  make  a  special  deter- 
mination of  it  as  described  above.     If  the  circle  is  graduated 
in  one  direction,  it  will  be  necessary  to  subtract  the  second 
reading  from  180°  and  then  take  the  mean  between  this  result 
and  the  first  altitude.     In  the  preceding  description  it  is  assumed 
that  the  plate  levels  remain  central  during  the  reversal  of  the 
instrument,  indicating  that  the  vertical  axis  is  truly  vertical. 
If  this  is  not  the  case,  the  instrument  should  be  relevelled  before 
the  second  altitude  is  measured,  the  difference  in  the  two  altitude 
readings  in  this  case  including  all  three  errors.     If  it  is  not  de- 
sirable to  relevel,  the  error  of  inclination  of  the  vertical  axis  may 


DESCRIPTION  OF  INSTRUMENTS  85 

still  be  eliminated  by  reading  the  vernier  of  the  vertical  circle 
in  each  of  the  two  positions  when  the  telescope  bubble  is  central, 
and  applying  these  corrections  separately.  With  an  instru- 
ment provided  with  a  vertical  arc  only  it  is  essential  that  the  axis 
of  the  telescope  bubble  be  made  parallel  to  the  line  of  sight,  and 
that  the  vertical  axis  be  made  truly  vertical.  To  make  the  axis 
vertical  without  adjusting  the  levels  themselves,  bring  both 
bubbles  to  the  centres  of  their  tubes,  turn  the  instrument  180° 
in  azimuth,  and  then  bring  each  bubble  half  way  back  to  the 
centre  by  means  of  the  levelling  screws.  When  the  axis  is  truly 
vertical,  each  bubble  should  remain  in  the  same  part  of  its  tube 
in  all  azimuths.  The  axis  may  always  be  made  vertical  by 
means  of  the  long  bubble  on  the  telescope;  this  is  done  by  set- 
ting it  over  one  pair  of  levelling  screws  and  centring  it  by  means 
of  the  tangent  screw  on  the  standard;  the  telescope  is  then 
revolved  about  the  vertical  axis,  and  if  the  bubble  moves  from 
the  centre  of  its  tube  it  is  brought  half  way  back  by  means  of 
the  tangent  screw,  and  then  centred  by  means  of  the  levelling 
screws.  This  process  should  be  repeated  to  test  the  accuracy 
of  the  levelling;  the  telescope  is  then  turned  at  right  angles 
to  the  first  position  and  the  whole  process  repeated.  This 
method  should  always  be  used  when  the  greatest  precision  is 
desired,  because  the  telescope  bubble  is  much  more  sensitive 
than  the  plate  bubbles. 

If  the  line  of  sight  is  not  at  right  angles  to  the  horizontal  axis, 
or  if  the  horizontal  axis  is  not  perpendicular  to  the  vertical  axis, 
the  errors  due  to  these  two  causes  maybe  eliminated  by  com- 
bining two  sets  of  measurements,  one  in  each  position  of  the 
instrument.  If  a  horizontal  angle  is  measured  with  the  vertical 
circle  on  the  observer's  right,  and  the  same  angle  again  observed 
with  the  circle  on  his  left,  the  mean  of  these  two  angles  is  free 
from  both  these  errors,  because  the  two  positions  of  the  horizontal 
axis  are  placed  symmetrically  about  a  true  horizontal  line,*  and 

*  Strictly  speaking,  they  are  placed  symmetrically  about  a  perpendicular  to 
the  vertical  axis. 


86  PRACTICAL  ASTRONOMY 

the  two  directions  of  the  sight  line  are  situated  symmetrically 
about  a  true  perpendicular  to  the  rotation  axis  of  the  telescope. 
If  the  horizontal  axis  is  not  perpendicular  to  the  vertical  axis  the 
line  of  sight  describes  a  plane  which  is  inclined  to  the  true  vertical 
plane.  In  this  case  the  sight  line  will  not  pass  through  the  zenith, 
and  both  horizontal  and  vertical  angles  will  be  in  error.  In 
instruments  intended  for  precise  work  a  striding  level  is  provided, 
which  may  be  set  on  the  pivots  of  the  horizontal  axis.  This 
enables  the  observer  to  level  the  axis  or  to  measure  its  inclina- 
tion without  reference  to  the  plate  bubbles.  The  striding  level 
should  be  used  in  both  the  direct  and  the  reversed  position  and 
the  mean  of  the  two  results  used  in  order  to  eliminate  the  errors 
of  adjustment  of  the  striding  level  itself.  If  the  line  of  sight  is 
not  perpendicular  to  the  horizontal  axis  it  will  describe  a  cone 
whose  axis  is  the  horizontal  axis  of  the  instrument.  The  line 
of  sight  will  in  general  not  pass  through  the  zenith,  even  though 
the  horizontal  axis  be  in  perfect  adjustment.  The  instrument 
must  either  be  used  in  two  positions,  or  else  the  cross  hairs  must 
be  adjusted.  Except  in  large  transits  it  is  not  usually  practicable 
to  determine  the  amount  of  the  error  and  allow  for  it. 

49.   Attachments  to  the  Engineer's  Transit.  —  Reflector. 

When  making  star  observations  with  the  transit  it  is  necessary 
to  make  some  arrangement  for  illuminating  the  field  of  view. 
Some  transits  are  provided  with  a  special  shade  tube  into  which 
is  fitted  a  mirror  set  at  an  angle  of  45°  and  with  the  central 
portion  removed.  By  means  of  a  lantern  held  at  one  side  of 
the  telescope  light  is  reflected  down  the  tube.  The  cross  hairs 
appear  as  dark  lines  against  the  bright  field.  The  stars  can  be 
seen  through  the  opening  in  the  centre  of  the  mirror.  If  no 
special  shade  tube  is  provided,  it  is  a  simple  matter  to  make  a 
substitute,  either  from  a  piece  of  bright  tin  or  by  fastening  a 
piece  of  tracing  cloth  or  oiled  paper  over  the  objective.  A  hole 
about  |  inch  in  diameter  should  be  cut  out,  so  that  the  light  from 
the  star  may  enter  the  lens.  If  cloth  or  paper  is  used,  the  lan- 
tern must  be  held  so  that  the  light  is  diffused  in  such  a  way  as 


DESCRIPTION  OF  INSTRUMENTS  87 

to  render  the  cross  hairs  visible.     The  light  should  be  held  so  as 
not  to  shine  into  the  observer's  eyes. 

50.  Prismatic  Eyepiece. 

When  altitudes  greater  than  about  55°  to  60°  are  to  be  meas- 
ured, it  is  necessary  to  attach  to  the  eyepiece  a  totally  reflecting 
prism  which  reflects  the  rays  at  right  angles  to  the  sight  line. 
By  means  of  this  attachment  altitudes  as  great  as  75°  can  be 
measured.  In  making  observations  on  the  sun  it  must  be 
remembered  that  the  prism  inverts  the  image,  so  that  with  a 
transit  having  an  erecting  eyepiece  with  the  prism  attached  the 
apparent  lower  limb  is  the  true  upper  limb;  the  positions  of  the 
right  and  left  limbs  are  not  affected  by  the  prism. 

51.  Sun  Glass. 

In  making  observations  on  the  sun  it  is  necessary  to  cover  the 
eyepiece  with  a  piece  of  dark  glass  to  protect  the  eye  from  the 
sunlight  while  observing.  The  sun  glass  should  not  be  placed 
in  front  of  the  objective.  If  no  shade  is  provided  with  the 
instrument,  sun  observations  may  be  made  by  holding  a  piece 
of  paper  behind  the  eyepiece  so  that  the  sun's  image  is  thrown 
upon  it.  By  drawing  out  the  eyepiece  tube  and  varying  the 
distance  at  which  the  paper  is  held,  the  images  of  the  sun  and 
the  cross  hairs  may  be  sharply  focussed.  By  means  of  this 
device  an  observation  may  be  quite  accurately  made  after  a 
little  practice. 

52.  The  Portable  Astronomical  Transit. 

The  astronomical  transit  differs  from  the  surveyor's  transit  chiefly  in  size  and 
in  the  manner  of  support.  The  diameter  of  the  object  glass  may  be  anywhere 
from  2  to  4  inches,  and  the  focal  length  from  24  to  48  inches.  The  instrument  is 
set  upon  a  stone  or  brick  pier.  The  cross  hairs  usually  consist  of  several  vertical 
hairs  (say  n  or  more)  instead  of  a  single  one  as  in  the  surveyor's  transit.  The 
motion  in  altitude  is  controlled  by  means  of  a  clamp  and  a  tangent  screw.  The 
azimuth  motion  is  usually  very  small,  simply  enough  to  allow  adjustments  to  be 
made,  as  the  transit  is  not  used  for  measuring  horizontal  angles.  The  axis  is 
levelled  or  its  inclination  measured  by  means  of  a  sensitive  striding  level. 

On  account  of  the  high  precision  of  the  work  done  with  the  astronomical  transit 
the  various  errors  have  to  be  determined  with  great  accuracy,  and  corresponding 
corrections  applied  to  the  observed  results.  The  transit  is  chiefly  used  in  the  plane 


PRACTICAL  ASTRONOMY 


of  the  meridian  for  determining  the  times  of  transit  of  stars.  The  principal  errors 
determined  and  allowed  for  are  (i)  azimuth,  or  deviation  from  the  true  meridian; 
(2)  inclination  of  the  horizontal  axis;  (3)  collimation,  or  deviation  of  the  sight  line 
from  the  true  perpendicular  to  the  rotation  axis.  The  corrections  to  reduce  an 
observed  time  to  the  true  time  of  transit  across  the  meridian  are  given  by  formulae 
[66]  to  [68].  These  corrections  would  apply  equally  well  to  observations  with  the 
engineer's  transit,  and  serve  to  show  the  relative  magnitudes  of  the  errors  for 
different  positions  of  the  objects  observed.  . 

[66] 


Azimuth  correction  =  a  cos  h  sec  D, 
Level  correction  =  b  sin  h  sec  D, 
Collimation  correction  =  c  sec  Z>, 


[67] 
[68] 


where  a,  b  and  c  are  the  errors  in  azimuth,  inclination  and  collimation  respectively 
(expressed  in  seconds  of  time),  and  h  is  the  altitude  and  D  the  declination  of  the 
star  observed.  From  these  formulas  Table  B  has  been  computed.  It  is  assumed 
that  the  instrument  is  i',  or  4s,  out  of  the  meridian  (a  =  4*);  that  the  axis  is 
inclined  i',  or  4s,  to  the  horizon  (b  —  4s);  and  that  the  sight  line  denned  by  the 
middle  (or  the  mean)  wire  is  i',  or  4s,  to  the  right  or  left  of  its  true  position  (c=4s). 
The  numbers  in  the  table  show  the  effect  of  these  errors  at  different  altitudes  and 
declinations. 

TABLE  B.     ERROR   IN    OBSERVED   TIME   OF   TRANSIT    (IN 
SECONDS    OF   TIME)    WHERE   a,  b   OR   c  =  i'. 


Declinations. 

I 

h 

0° 

10° 

20° 

30° 

40° 

50° 

60° 

70° 

80° 

h 

<£ 

c 

W 

0° 

o*.o 

os.o 

0s.  0 

os.o 

os.o 

0s.  0 

0s.  0 

0s.  0 

o*.o 

90° 

§ 

W 

.2 

10 

0.7 

0.7 

0.8 

0.8 

0.9 

I  .  I 

1.4 

2  .0 

4.0 

80 

1 

20 

1.4 

1.4 

1.4 

1.6 

1.8 

2.  I 

2.7 

4.0 

7-9 

70 

| 

1 

3° 

2.0 

2.0 

2.  I 

2-3 

2.6 

3-1 

4.0 

5-8 

"•S 

60 

'N 

g 

40 

2.6 

2.6 

2.7 

3-o 

3-4 

4.0 

S-2 

7-5 

14.8 

5° 

M 

u 

5° 

3.1 

3-1 

3-3 

3-6 

4.0 

4.8 

6.1 

9.0 

17.6 

40 

•8 

1 

60 

3-5 

3-5 

3-7 

4.0 

4-5 

5-4 

6.9 

10.  I 

19.9 

3° 

3 

'1 

70 

3-8 

3-8 

4.0 

4.4 

4.9 

5-8 

7-5 

II.  0 

21.6 

20 

3 

80 

3-9 

4.0 

4.2 

4.6 

S-2 

6.1 

7-9 

"•5 

22.7 

10 

90 

4.0 

4.1 

4.2 

4.6 

5-2 

6.2 

8.0 

11.7 

23.0 

0 

Note.  —  Use  the  bottom  line  for  the  collimation  error. 

53.   The  Sextant. 

The  sextant  is  an  instrument  for  measuring  the  angular  dis- 
tance between  two  objects,  the  angle  always  lying  in  the  plane 


DESCRIPTION  OF   INSTRUMENTS  89 

through  the  two  objects  and  the  eye  of  the  observer.  It  is 
particularly  useful  at  sea  because  it  does  not  require  a  steady 
support  like  the  transit.  It  consists  of  a  frame  carrying  a 
graduated  arc,  A  B,  Fig.  43,  about  60°  long,  and  two  mirrors  / 
and  H,  the  first  one  movable,  the  second  one  fixed.  At  the 
centre  of  the  arc,  7,  is  a  pivot  on  which  swings  an  arm  7F,  6  to 
8  inches  long.  This  arm  carries  a  vernier  V  for  reading  the 


FIG.  43 

angles  on  the  arc  AB.  Upon  this  arm  is  placed  the  index  glass 
/.  At  H  is  the  horizon  glass.  Both  of  these  mirrors  are  set 
so  that  their  planes  are  perpendicular  to  the  plane  of  the  arc 
AB,  and  so  that  when  the  vernier  reads  o°  the  mirrors  are  parallel. 
The  half  of  the  mirror  H  which  is  farthest  from  the  frame  is 
unsilvered,  so  that  objects  may  be  viewed  directly  through  the 
glass.  In  the  silvered  portion  other  objects  may  be  seen  by 
reflection  from  the  mirror  /  to  the  mirror  H  and  thence  to 
point  0.  At  a  point  near  0  (on  the  line  HO)  is  a  telescope  of 
low  power  for  viewing  the  objects.  Between  the  two  mirrors 


9°  PRACTICAL  ASTRONOMY 

and  also  to  the  left  of  H  are  colored  shade  glasses  to  be  used  when 
making  observations  on  the  sun.  The  principle  of  the  instru- 
ment is  as  follows :  —  A  ray  of  light  coming  from  an  object  at 
C  is  reflected  by  the  mirror  /  to  H,  where  it  is  again  reflected 
to  O.  The  observer  sees  the  image  of  C  in  apparent  coincidence 
with  the  object  at  D.  The  arc  is  so  graduated  that  the  reading 
of  the  vernier  gives  directly  the  angle  between  OC  and  OD. 
Drawing  the  perpendiculars  FE  and  HE  to  the  planes  of  the 
two  mirrors,  it  is  seen  that  the  angle  between  the  mirrors  is 
a.  —  |8.  Prolonging  CI  and  DH  to  meet  at  O,  it  is  seen  that  the 
angle  between  the  two  objects  is  2  a  —  2  ft.  The  angle  between 
the  mirrors  is  therefore  half  the  angle  between  the  objects  that 
appear  to  coincide.  In  order  that  the  true  angle  may  be  read 
directly  from  the  arc  each  half  degree  is  numbered  as  though  it 
were  a  degree.  It  will  be  seen  that  the  position  of  the  vertex  O 
is  variable,  but  since  all  objects  observed  are  at  great  distances 
the  errors  caused  by  changes  in  the  position  of  O  are  always 
negligible  in  astronomical  observations. 

The  sextant  is  in  adjustment  when,  (i)  both  mirrors  are  per- 
pendicular to  the  plane  of  the  arc;  (2)  the  line  of  sight  of  the 
telescope  is  parallel  to  the  plane  of  the  arc;  and  (3)  the  vernier 
reads  o°  when  the  mirrors  are  parallel  to  each  other.  If  the 
vernier  does  not  read  o°  when  the  doubly  reflected  image  of  a 
point  coincides  with  the  object  as  seen  directly,  the  index  cor- 
rection may  be  determined  and  applied  as  follows.  Set  the 
vernier  to  read  about  30'  and  place  the  shades  in  position  for 
sun  observations.  When  the  sun  is  sighted  through  the  tele- 
scope two  images  will  be  seen  with  their  edges  nearly  in  contact. 
This  contact  should  be  made  as  nearly  perfect  as  possible  and 
the  vernier  reading  recorded.  This  should  be  repeated  several 
times  to  increase  the  accuracy.  Then  set  the  vernier  about  30' 
on  the  opposite  side  of  the  zero  point  and  repeat  the  whole 
operation,  the  reflected  image  of  the  sun  now  being  on  the 
opposite  side  of  the  direct  image.  If  the  shade  glasses  are  of 
different  colors  the  contacts  can  be  more  precisely  made.  Half 


DESCRIPTION  OF  INSTRUMENTS  91 

the  difference  of  the  two  (average)  readings  is  the  index  correc- 
tion. If  the  reading  of  the  arc  was  the  greater,  the  correction 
is  to  be  added  to  all  readings  of  the  vernier ;  if  the  greater  reading 
was  on  the  arc,  the  correction  must  be  subtracted. 

In  measuring  an  altitude  of  the  sun  above  the  sea  horizon  the 
observer  directs  the  telescope  to  the  point  on  the  horizon  ver- 
tically under  the  sun  and  then  moves  the  index  arm  until  the 
reflected  image  of  the  sun  comes  into  view.  The  sea  horizon 
can  be  seen  through  the  plain  glass  and  the  sun  is  seen  in  the 
mirror.  The  sun's  lower  limb  is  then  set  in  contact  with  the 
horizon  line.  In  order  to  be  certain  that  the  angle  is  measured 
to  the  point  vertically  beneath  the  sun,  the  instrument  is  tipped 
slowly  right  and  left,  causing  the  sun's  image  to  describe  an  arc. 
This  arc  should  be  just  tangent  to  the  horizon.  If  at  any  point 
the  sun's  limb  goes  below  the  horizon  the  altitude  measured  is 
too  great.  The  vernier  reading  corrected  for  index  error  and 
dip  is  the  apparent  altitude  of  the  lower  limb  above  the  true 
horizon. 

54.   Artificial  Horizon. 

When  altitudes  are  to  be  measured  on  land  the  visible  horizon 
cannot  be  used,  and  the  artificial  horizon  must  be  used  instead. 
The  surface  of  any  heavy  liquid,  like  mercury,  molasses,  or 
heavy  oil,  may  be  used  for  this  purpose.  When  the  liquid  is 
placed  in  a  basin  and  allowed  to  come  to  rest,  the  surface  is 
perfectly  level,  and  in  this  surface  the  reflected  image  of  the  sun 
may  be  seen,  the  image  appearing  as  far  below  the  horizon  as 
the  sun  is  above  it.  Another  convenient  form  of  horizon  con- 
sists of  a  piece  of  black  glass,  with  plane  surfaces,  mounted  on  a 
frame  supported  by  levelling  screws.  This  horizon  is  brought 
into  position  by  placing  a  spirit  level  on  the  glass  surface  and 
levelling  alternately  in  two  positions  at  right  angles  to  each 
other.  This  form  of  horizon  is  not  as  accurate  as  the  mercury 
surface  but  is  often  more  convenient.  The  principle  of  the 
artificial  horizon  may  be  seen  from  Fig.  44.  Since  the  image 
seen  in  the  horizon  is  as  far  below  the  true  horizon  as  the  sun  is 


92  PRACTICAL  ASTRONOMY 

above  it,  the  angle  between  the  two  is  2  h.  In  measuring  this 
angle  the  observer  points  his  telescope  toward  the  artificial 
horizon  and  then  brings  the  reflected  sun  down  into  the  field  of 
view  by  means  of  the  index  arm.  By  placing  the  apparent 
lower  limb  of  the  reflected  sun  in  contact  with  the  apparent 
upper  limb  of  the  image  seen  in  the  mercury  surface,  the  angle 
measured  is  twice  the  altitude  of  the  sun's  lower  limb.  The  two 
points  in  contact  are  really  images  of  the  same  point.  If  the 
telescope  inverts  the  image,  this  statement  applies  to  the  upper 
limb.  The  index  correction  must  be  applied  before  the  angle  is 


Sextant- 


FIG.  44 

divided  by  2  to  obtain  the  altitude.  In  using  the  mercury  hori- 
zon care  must  be  taken  to  protect  it  from  the  wind,  otherwise 
small  waves  on  the  mercury  surface  will  blur  and  distort  the 
image.  The  horizon  is  usually  provided  with  a  roof-shaped 
cover  having  glass  windows,  but  unless  the  glass  has  parallel 
faces  this  introduces  an  error  into  the  result.  A  good  substitute 
for  the  glass  cover  is  one  made  of  fine  mosquito  netting.  This 
will  break  the  force  of  the  wind  if  it  is  not  blowing  hard,  and 
does  not  introduce  errors  into  the  measurement. 

55.   Chronometer. 

The  chronometer  is  simply  an  accurately  constructed  watch 
with  a  special  form  of  escapement.  Chronometers  may  be 


DESCRIPTION  OF  INSTRUMENTS  93 

regulated  for  either  sidereal  or  mean  time.     The  beat  is  usually 
a  half  second.     Those  designed  to  register  the  time  on  chrono- 
graphs are  arranged  to  break  an  electric  circuit  at  the  end  of 
every  second  or  every  two  seconds.     The  60 th  second  is  dis- 
tinguished either  by  the  omission  of  the  break  at  the  previous 
second,  or  by  an  extra  break,  according  to  the  construction  of  the 
chronometer.     Chronometers  are  usually  hung  in  gimbals   to 
keep  them  level  at  all  times;  this  is  invariably  done  when  they 
are  taken  to  sea.     It  is  important  that  the  temperature  of  the 
chronometer  should  be  kept  as  nearly  uniform  as  possible,  be- 
cause fluctuation  in  temperature  is  the  greatest  source  of  error. 
Two  chronometers  of  the  same  kind  cannot  be  directly  com- 
pared with  great  accuracy,  os.i  or  os.2  being  about  as  close  as 
the  difference  can  be  estimated.     But  a  sidereal  and  a  solar  chro- 
nometer can  easily  be  compared  within  a  few  hundredths  of  a 
second.     On  account  of  the  gain  of  the  sidereal  on  the  solar 
chronometer,  the  beats  of  the  two  will  coincide  once  in  about 
every  3  m  03*.     If  the  two  are  compared  at  the  instant  when  the 
beats  are  apparently  coincident,  then  it  is  only  necessary  to 
note  the  seconds  and  half  seconds,  as  there  are  no  fractions  to 
be  estimated.     By  making    several    comparisons  and  reducing 
them  to  some  common  instant  of  time  it  is  readily  seen  that 
the  comparison  is  correct  within  a  few  hundredths  of  a  second. 
The  accuracy  of  the  comparison  depends  upon  the  fact  that  the 
ear  can  detect  a  much  smaller  interval  between  the  two  beats 
than  can  possibly  be  estimated  when  comparing  two  chronome- 
ters whose  beats  do  not  coincide. 

56.    Chronograph. 

The  chronograph  is  an  instrument  for  recording  the  time  kept  by  a  chronometer 
and  also  any  observations  the  times  of  which  it  is  desired  to  determine.  A  piece 
of  paper  is  wrapped  about  a  cylinder,  which  is  revolved  by  a  mechanism  at  a  uniform 
rate.  A  pen  in  contact  with  the  paper  is  held  on  an  arm,  connected  with  the  arma- 
ture of  an  electro-magnet,  in  such  a  way  that  the  pen  draws  a  continuous  line  which 
has  notches  in  it  corresponding  to  the  breaks  in  the  circuit  made  by  the  chro- 
nometer. By  means  of  this  instrument  the  time  is  represented  accurately  on  the 
sheet  as  a  linear  distance.  If  it  is  desired  to  record  the  instant  when  any  event 


94 


PRACTICAL  ASTRONOMY 


occurs,  such  as  the  passage  of  a  star  over  a  cross  hair,  the  observer  presses  a  tele- 
graph key  which  breaks  the  same  circuit,  and  a  mark  is  made  on  the  chronograph 
sheet.  The  instant  of  the  observation  may  be  scaled  from  the  record  sheet  with 
great  precision. 

57.   The  Zenith  Telescope. 

The  zenith  telescope  is  an  instrument  designed  for  making  observations  for 
latitude  by  a  special  method  devised  by  Capt.  Andrew  Talcott,  and  which  bears 
his  name.  The  instrument  consists  of  a  telescope  having  a  vertical  and  a  horizontal 
axis  like  the  transit;  the  telescope  is  attached  to  one  end  of  the  horizontal  axis  in- 
stead of  at  the  centre.  The  essential  features  of  the  instrument  are  (i)  a  microm- 
eter, placed  in  the  focus  of  the  eyepiece,  for 
measuring  small  differences  in  zenith  distance, 
and  (2)  a  sensitive  spirit  level,  attached  to  a 
small  vertical  circle  on  the  telescope  tube,  for 
measuring  small  deflections  of  the  vertical  axis. 
The  telescope  is  used  in  the  plane  of  the 
meridian.  There  are  two  stops  whose  positions 
can  be  regulated  so  that  the  telescope  may  be 
quickly  shifted,  by  a  rotation  about  the  ver- 
tical axis,  from  the  north  meridian  to  the 
south  meridian.  The  observation  consists  in 
measuring  with  the  micrometer  the  difference 
in  zenith  distance  of  two  stars,  one  north  of 
the  zenith  and  one  south,  which  culminate 
within  a  few  minutes  of  each  other,  and  in 
taking  readings  of  the  spirit  level  at  the  same 
time  the  micrometer  settings  are  made.  A 
FIG.  45.  THE  ZENITH  TELESCOPE  diagram  of  the  instrument  in  the  two  posi- 
tions is  given  in  Fig.  45.  The  inclination  of 

the  telescope  to  the  vertical  is  not  changed  between  the  two  observations,  so  it  is 
essential  that  the  zenith  distances  of  the  two  stars  should  be  so  nearly  equal  that 
both  will  come  within  the  range  of  the  micrometer  screw,  usually  30'  or  less. 
The  principle  involved  in  this  method  may  be  seen  from  Fig.  46.  From  the 
observed  zenith  distance  of  the  star  5s  the  latitude  is 


and  from  the  star  Sr 


Taking  the  mean, 


L  =  Ds  +  zs 

L  =  Dn  -  zn. 


[69] 


The  latitude  is  therefore  the  mean  of  the  declinations  corrected  by  half  the  differ- 
ence of  the  zenith  distances.  The  declination  may  be  computed  from  the  star 
catalogues,  and  the  difference  in  zenith  distance  may  be  very  accurately  measured 
with  the  micrometer  screw.  It  is  evidently  essential  that  the  telescope  should 


DESCRIPTION   OF   INSTRUMENTS 


95 


have  the  same  inclination  to  the  vertical  in  each  case.  If  the  inclination  changes, 
however,  the  amount  of  this  change  is  accurately  determined  from  the  level  readings 
already  mentioned  (see  Art.  70). 


FIG.  46 

58.   Suggestions  about  Observing. 

The  instrument  used  for  making  such  observations  as  are 
described  in  this  book  will  usually  be  either  the  engineer's  transit 
or  the  sextant.  In  using  the  transit  care  must  be  taken  to  give 
the  tripod  a  firm  support.  It  is  well  to  set  the  transit  in  position 
some  time  before  the  observations  are  to  be  begun ;  this  allows 
the  instrument  to  assume  the  temperature  of  the  air  and  the 
tripod  legs  to  come  to  a  firm  bearing  on  the  ground.  The 
observer  should  handle  the  instrument  with  great  care,  par- 
ticularly during  night  observations,  when  the  instrument  is 
likely  to  be  accidentally  disturbed.  In  reading  angles  at  night 
it  is  important  to  hold  the  light  in  such  a  position  that  the 
graduations  on  the  circle  are  plainly  visible  and  may  be  viewed 
along  the  lines  of  graduation,  not  obliquely.  By  changing  the 
position  of  the  lantern  and  the  position  of  the  eye  it  will  be 
found  that  the  reading  varies  by  larger  amounts  than  would  be 
expected  when  reading  in  the  daylight.  Care  should  be  taken 
not  to  touch  the  graduated  silver  circles,  as  they  soon  become 
tarnished.  The  lantern  should  be  held  so  as  to  heat  the  instru- 
ment as  little  as  possible,  and  so  as  not  to  shine  into  the  observer's 
eyes.  Time  may  be  saved  and  mistakes  avoided  if  the  program 
of  observations  is  laid  out  beforehand,  so  that  the  observer  knows 
just  what  is  to  be  done  and  the  proper  order  of  the  different 


96 


PRACTICAL  ASTRONOMY 


steps.  The  observations  should  be  arranged  so  as  to  eliminate 
instrumental  errors,  usually  by  means  of  reversals;  but  if  this 
is  not  practicable,  then  the  instrument  must  be  put  in  good 
adjustment.  The  index  correction  should  be  determined  and 
applied,  unless  it  can  be  eliminated  by  the  method  of  observing. 
In  observations  for  time  it  will  often  be  necessary  to  use  an 
ordinary  watch.  If  there  are  two  observers,  one  can  read  the 
time  while  the  other  makes  the  observations.  If  a  chronometer 
is  used,  one  observer  may  easily  do  the  work  of  both,  and  at  the 

z 


FIG.  47 

same  time  increase  the  accuracy.  In  making  observations  by 
this  method  (called  the  "  eye  and  ear  method  ")  the  observer 
looks  at  the  chronometer,  notes  the  reading  at  some  instant,  say 
at  the  beginning  of  some  minute,  and,  listening  to  the  half-second 
beats,  carries  along  the  count  mentally  and  without  looking  at 
the  chronometer.  In  this  way  he  can  note  the  second  and 
estimate  the  fraction  without  taking  his  attention  from  the  star 
and  cross  hair.  After  making  his  observation  he  may  check  his 
count  by  again  looking  at  the  chronometer  to  see  if  the  two 
agree.  After  a  little  practice  this  method  can  be  used  easily 
and  accurately.  In  using  a  watch  it  is  possible  for  one  observer 
to  make  the  observations  and  also  note  the  time,  but  it  cannot 
be  done  with  any  such  precision  as  with  the  chronometer,  be- 
cause on  account  of  the  rapidity  of  the  ticks  (5  per  second), 
the  observer  cannot  count  the  seconds  mentally.  The  observer 


DESCRIPTION  OF  INSTRUMENTS 


97 


must  in  this  case  look  quickly  at  his  watch  and  make  an  allow- 
ance, if  it  appears  necessary,  for  the  time  lost  in  looking  up  and 

taking  the  reading. 

Problems 

i.  Show  that  if  the  sight  line  makes  an  angle  c  with  the  perpendicular  to  the 
horizontal  axis  (Fig.  47)  the  horizontal  angle  between  two  points  is  in  error  by 
the  angle 

c  sec  hf  —  c  sec  h", 

where  h'  and  h"  are  the  altitudes  of  the  two  points. 


I  tan  h 


FIG.  48a. 

2.  Show  that  if  the  horizontal  axis  is  inclined  to  the  horizon  by  the  angle  i 
(Figs.  48a  and  48b)  the  effect  upon  the  azimuth  of  the  sight  line  is  i  tan  A,  and  that 
an  angle  is  in  error  by 

i  (tan  h'  —  tan  h"), 
where  h'  and  h"  are  the  altitudes  of  the  points. 


CHAPTER  IX 
THE    CONSTELLATIONS 

59.  The  Constellations. 

A  study  of  the  constellations  is  not  really  a  part  of  the  subject 
of  Practical  Astronomy,  and  in  much  of  the  routine  work  of 
observing  it  would  be  of  comparatively  little  value,  since  the 
stars  used  can  be  identified  by  means  of  their  coordinates  and  a 
knowledge  of  their  positions  in  the  constellations  is  not  essential. 
If  an  observer  has  placed  his  transit  in  the  meridian  and  knqws 
approximately  his  latitude  and  the  local  time,  he  can  identify 
stars  crossing  the  meridian  by  means  of  the  times  and  the  alti- 
tudes at  which  they  culminate.  But  in  making  occasional 
observations  with  small  instruments,  and  where  much  of  the 
astronomical  data  is  not  known  to  the  observer  at  the  time,  some 
knowledge  of  the  stars  is  necessary.  When  a  surveyor  is  be- 
ginning a  series  of  observations  in  a  new  place  and  has  no  accu- 
rate knowledge  of  his  position  nor  the  position  of  the  celestial 
sphere  at  the  moment,  he  must  be  able  to  identify  certain  stars 
in  order  to  make  approximate  determinations  of  the  quantities 
sought. 

60.  Method  of  Naming  Stars. 

The  whole  sky  is  divided  in  an  arbitrary  manner  into  irregular 
areas,  all  of  the  stars  in  any  one  area  being  called  a  constellation 
and  given  a  special  name.  The  individual  stars  in  any  constel- 
lation are  usually  distinguished  by  a  name,  a  Greek  letter,*  or 
a  number.  The  letters  are  usually  assigned  in  the  order  of 
brightness  of  the  stars,  a  being  the  brightest,  0  the  next,  and  so 
on.  A  star  is  named  by  stating  first  its  letter  and  then  the  name 
of  the  constellation  in  the  (Latin)  genitive  form.  For  instance, 

*  The  Greek  alphabet  is  given  on  p.  190. 


THE   CONSTELLATIONS  99 

in  the  constellation  Ursa  Minor  the  star  a  is  called  a  Ursa 
Minoris;  the  star  Vega  in  the  constellation  Lyra  is  called 
a  Lyra.  When  two  stars  are  very  close  together  and  have 
been  given  the  same  letter,  they  are  often  distinguished  by  the 
numbers  i,  2,  etc.,  written  above  the  letter,  as,  for  example, 
a2  Capricorni,  meaning  that  the  star  passes  the  meridian  after 
a1  Capricorni. 

61.  Magnitudes. 

The  brightness  of  stars  is  shown  on  a  numerical  scale  by  their 
magnitudes.  A  star  having  a  magnitude  i  is  brighter  than  one 
having  a  magnitude  2.  On  the  scale  of  magnitudes  in  use  a  few 
of  the  brightest  stars  have  fractional  or  negative  magnitudes. 
Stars  of  the  fifth  magnitude  are  visible  to  the  naked  eye  only 
under  favorable  conditions.  Below  the  fifth  magnitude  a  tele- 
scope is  usually  necessary  to  render  the  star  visible. 

62.  Constellations  Near  the  Pole. 

The  stars  of  the  greatest  importance  to  the  surveyor  are  those 
near  the  pole.  In  the  northern  hemisphere  the  pole  is  marked 
by  a  second-magnitude  star,  called  the  polestar,  Polaris,  or 
a  Ursce  Minoris,  which  is  about  i°  08'  distant  from  the  pole 
at  the  present  time  (1916).  This  distance  is  now  decreasing 
at  the  rate  of  about  one-third  of  a  minute  per  year,  so  that  for 
several  centuries  this  star  will  be  close  to  the  celestial  north  pole. 
On  the  same  side  of  the  pole  as  Polaris,  but  much  farther  from 
it,  is  a  constellation  called  Cassiopeia,  the  five  brightest  stars 
of  which  form  a  rather  unsymmetrical  letter  W  (Fig.  49).  The 
lower  left-hand  star  of  this  constellation,  the  one  at  the  bottom 
of  the  first  stroke  of  the  W,  is  called  6,  and  is  of  importance  to 
the  surveyor  because  it  is  very  nearly  on  the  hour  circle  passing 
through  Polaris  and  the  pole;  in  other  words  its  right  ascension 
is  nearly  the  same  as  that  of  Polaris.  On  the  opposite  side  of 
the  pole  from  Cassiopeia  is  Ursa  Major,  or  the  great  dipper,  a 
rather  conspicuous  constellation.  The  star  f,  which  is  at  the 
bend  in  the  dipper  handle,  is  also  nearly  on  the  same  hour  circle 
as  Polaris  and  d  Cassiopeia.  If  a  line  be  drawn  on  the  sphere 


100  PRACTICAL  ASTRONOMY 

between  5  Cassiopeia  and  £  Ursa  Majoris,  it  will  pass  nearly 
through  Polaris  and  the  pole,  and  will  show  at  once  the  position 
of  Polaris  in  its  diurnal  circle.  The  two  stars  in  the  bowl  of 
the  great  dipper  on  the  side  farthest  from  the  handle  are  in  a 
line  which,  if  prolonged,  would  pass  near  to  Polaris.  These 
stars  are  therefore  called  the  pointers  and  may  be  used  to  find 
the  polestar.  There  is  no  other  star  near  Polaris  which  is 
likely  to  be  confused  with  it.  Another  star  which  should  be 
remembered  is  /8  Cassiopeia,  the  one  at  the  upper  right-hand 
corner  of  the  W.  Its  right  ascension  is  very  nearly  oh  and 
therefore  the  hour  circle  through  it  passes  nearly  through  the 
equinox.  It  is  possible  then,  by  simply  glancing  at  (3  Cassiopeia 
and  the  polestar,  to  estimate  approximately  the  local  sidereal 
time.  When  /5  Cassiopeia  is  vertically  above  the  polestar  it 
is  nearly  oh  sidereal  time;  when  the  star  is  below  the  polestar 
it  is  i2h  sidereal  time;  half  way  between  these  positions,  left  and 
right,  it  is  6h  and  18*,  respectively.  In  intermediate  positions 
the  hour  angle  of  the  star  (  =  sidereal  time)  may  be  roughly 
estimated. 

63.   Constellations  Near  the  Equator. 

The  principal  constellations  within  45°  of  the  equator  are 
shown  in  Figs.  50  to  52.  Hour  circles  are  drawn  for  each  hour 
of  R.  A.  and  parallels  for  each  10°  of  declination.  The  approxi- 
mate decimation  and  right  ascension  of  a  star  may  be  obtained 
by  scaling  the  coordinates  from  the  chart.  The  position  of  the 
ecliptic,  or  sun's  path  in  the  sky,  is  shown  as  a  curved  line.  The 
moon  and  the  planets  are  always  found  near  this  circle  because 
the  planes  of  their  orbits  have  only  a  small  inclination  to  the 
earth's  orbit.  A  belt  extending  about  8°  each  side  of  the  ecliptic 
is  called  the  Zodiac,  and  all  the  members  of  the  solar  system 
will  always  be  found  within  this  belt.  The  constellations  along 
this  belt,  and  which  have  given  the  names  to  the  twelve  "  signs 
of  the  Zodiac,"  are  Aries,  Taurus,  Gemini,  Cancer,  Leo,  Virgo, 
Libra,  Scorpio,  Sagittarius,  Capricornus,  Aquarius,  and  Pisces. 
These  constellations  were  named  many  centuries  ago,  and  the 


FlG.   49.      CONSTELLATIC* 


MAPI 


BOUT  THE  NORTH  POLE 


Brighte 


Scale  of  Magnitude 
er  than  1.5       1.5^.9         2.0-2.9        3.0-3.9        4  or  fainte 


MARCH 


FEBRUARY 


40 


FIG.  50.    PRINCIPAL  FIXED  STARS  BETWI 


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FIG.  51.    PRINCIPAL  FIXED  STARS  BETW: 


MAP  III 


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L.  UPDATES  CO.,  N.V. 


DECLINATIONS  45°  NORTH  AND  45°  SOUTH 


Scale  of  Magnitude 

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J&iffhter  thanlS    1.5-1.9     £0  2.9     3.0-3.9     1  or  fainter 

&  *          *      '      + 


FlG.    53.      CONSTELLATIO 


MAPV 


VBOUT  THE  SOUTH  POLE 


THE   CONSTELLATIONS'  '    IOI 

names  have  been  retained,  both  for  the  constellations  themselves 
and  also  for  the  positions  in  the  ecliptic  which  they  occupied  at 
that  time.  But  on  account  of  the  continuous  westward  motion 
of  the  equinox,  the  "  signs  "  no  longer  correspond  to  the  con- 
stellations of  the  same  name.  For  example,  the  sign  of  Aries 
extends  from  the  equinoctial  point  to  a  point  on  the  ecliptic 
30°  eastward,  but  the  constellation  actually  occupying  this 
space  at  present  is  Pisces.  In  Figs.  50  to  52  the  constellations 
are  shown  as  seen  by  an  observer  on  the  earth,  not  as  they  would 
appear  on  a  celestial  globe.  On  account  of  the  form  of  pro- 
jection used  in  these  maps  there  is  some  distortion,  but  if  the 
observer  faces  south  and  holds  the  page  up  at  an  altitude  equal 
to  his  co-latitude,  the  map  represents  the  constellations  very 
nearly  as  they  will  appear  to  him.  The  portion  of  the  map  to  be 
used  in  any  month  is  that  marked  with  the  name  of  the  month 
at  the  top;  for  example,  the  stars  under  the  word  "  February  " 
are  those  passing  the  meridian  in  the  middle  of  February  at 
about  9  P.M.  For  other  hours  in  the  evening  the  stars  on  the 
meridian  will  be  those  at  a  corresponding  distance  right  or  left, 
according  as  the  time  is  earlier  or  later  than  9  P.M.  The  approxi- 
mate right  ascension  of  a  point  on  the  meridian  may  be  found  at 
any  time  as  follows:  First  compute  the  R.  A.  of  the  sun  by 
allowing  2h  per  month,  or  more  nearly  4™  per  day  for  every 
day  since  March  23,  remembering  that  the  R.  A.  of  the  sun  is 
always  increasing.  Add  this  R.  A.  to  the  local  mean  time  and 
the  result  is  the  sidereal  time  or  right  ascension  of  a  star  on  the 
meridian. 

Example.  On  October  10  the  R.  A.  of  the  sun  is  6  X  2h  +  17  X 
4m  =  13*  o8m.  At  Qh  P.M.  (local  mean  time)  the  sidereal  time 
is  13*  o8w  +  Qh  oom  =  22^  o8m.  A  star  having  a  R.  A.  of  22*  08™ 
would  therefore  be  close  to  the  meridian  at  9  P.M. 

Fig-  53  shows  the  stars  about  the  south  celestial  pole.  There 
is  no  bright  star  near  the  south  pole,  so  that  the  convenient 
methods  of  determining  the  meridian  by  observations  on  the 
polestar  are  not  practicable  in  the  southern  hemisphere. 


PRACTICAL  ASTRONOMY 

64.    The  Planets. 

In  using  the  star  maps,  the  student  should  be  on  the  lookout 
for  planets.  These  cannot  be  placed  on  the  maps  because  their 
positions  are  rapidly  changing.  If  a  bright  star  is  seen  near  the 
ecliptic,  and  its  position  does  not  correspond  to  that  of  a  star 
on  the  map,  it  is  a  planet.  The  planet  Venus  is  very  bright  and 
is  never  very  far  east  or  west  of  the  sun;  it  will  therefore  be 
seen  a  little  before  sunrise  or  a  little  after  sunset.  Mars,  Jupi- 
ter, and  Saturn  move  in  orbits  which  are  outside  of  that  of  the 
earth  and  therefore  appear  to  us  to  make  a  complete  circuit  of 
the  heavens.  Mars  makes  one  revolution  around  the  sun  in 
i  year  10  months,  Jupiter  in  about  12  years,  and  Saturn  in 
29^  years.  Jupiter  is  the  brightest,  and  when  looked  at  through 
a  small  telescope  shows  a  disc  like  that  of  the  full  moon;  four 
satellites  can  usually  be  seen  lying  nearly  in  a  straight  line. 
Saturn  is  not  as  large  as  Jupiter,  but  in  a  telescope  of  moderate 
power  its  rings  can  be  distinguished;  in  a  low-power  telescope 
the  planet  appears  to  be  elliptical  in  form.  Mars  is  reddish  in 
color  and  shows  a  disc. 


CHAPTER  X 
OBSERVATIONS    FOR    LATITUDE 

IN  this  chapter  and  the  three  immediately  following  are  given 
the  more  common  methods  of  determining  latitude,  time,  longi- 
tude, and  azimuth  with  small  instruments.  Those  which  are 
simple  and  direct  are  printed  in  large  type,  and  may  be  used  for 
a  short  course  in  the  subject.  Following  these  are  given,  in 
smaller  type,  several  methods  which,  although  less  simple,  are  very 
useful  to  the  engineer;  these  methods  require  a  knowledge  of 
other  data  which  the  engineer  must  obtain  by  observation,  and 
are  therefore  better  adapted  to  a  more  extended  course  of  study. 

65.   Latitude  by  a  Circumpolar  Star  at  Culmination. 

This  method  may  be  used  with  any  circumpolar  star,  but 
Polaris  is  the  best  one  to  use,  when  it  is  practicable  to  do  so, 
because  it  is  of  the  second  magnitude,  while  all  of  the  other 
close  circumpolars  are  quite  faint.  The  observation  consists 
in  measuring  the  altitude  of  the  star  when  it  is  a  maximum  or  a 
minimum,  or,  in  other  words,  when  it  is  on  the  observer's  me- 
ridian. This  altitude  may  be  obtained  by  trial,  and  it  is  not 
necessary  to  know  the  exact  instant  when  the  star  is  on  the 
meridian.  The  approximate  time  when  the  star  is  at  culmina- 
tion may  be  obtained  from  Table  V  or  by  formulae  [39]  and  [49]. 
It  is  not  necessary  to  know  the  time  with  accuracy,  but  it  will 
save  unnecessary  waiting  if  the  time  is  known  approximately. 
In  the  absence  of  any  definite  knowledge  of  the  time  of  culmina- 
tion, the  position  of  the  pole  star  with  respect  to  the  meridian  may 
be  estimated  by  noting  the  positions  of  the  constellations.  When 
6  Cassiopeia  is  directly  above  or  below  Polaris  the  latter  is  at 
upper  or  lower  culmination.  The  observation  should  be  begun 
some  time  before  one  of  these  positions  is  reached.  The  hori- 

103 


104  PRACTICAL  ASTRONOMY 

zontal  cross  hair  of  the  transit  should  be  set  on  the  star*  and  the 
motion  of  the  star  followed  by  means  of  the  tangent  screw  of  the 
horizontal  axis.  When  the  desired  maximum  or  minimum  is 
reached  the  vertical  arc  is  read.  The  index  correction  should 
then  be  determined.  If  the  instrument  has  a  complete  vertical 
circle  and  the  time  of  culmination  is  known  approximately,  it 
will  be  well  to  eliminate  instrumental  errors  by  taking  a  second 
altitude  with  the  instrument  reversed,  provided  that  neither 
observation  is  made  more  than  4 m  or  5  m  from  the  time  of  culmi- 
nation. If  the  star  is  a  faint  one,  and  therefore  difficult  to  find, 
it  may  be  necessary  to  compute  its  approximate  altitude  (using 
the  best  known  value  for  the  latitude)  and  set  off  this  altitude 
on  the  vertical  arc.  The  star  may  be  found  by  moving  the 
telescope  slowly  right  and  left  until  the  star  comes  into  the  field 
of  view.  Polaris  can  usually  be  found  in  this  manner  some  time 
before  dark,  when  it  cannot  be  seen  with  the  unaided  eye.  It 
is  especially  important  to  focus  the  telescope  carefully  before 
attempting  to  find  the  star,  for  the  slightest  error  of  focus  may 
render  the  star  invisible.  The  focus  may  be  adjusted  by  look- 
ing at  a  distant  terrestrial  object  or,  better  still,  by  sighting  at 
the  moon  or  at  a  planet  if  one  is  visible.  If  observations  are  to 
be  made  frequently  with  a  surveyor's  transit,  it  is  well  to  have 
a  reference  mark  scratched  on  the  telescope  tube,  so  that  the 
objective  may  be  set  at  once  at  the  proper  focus. 

The  latitude  is  computed  from  Equa.  [3]  or  [4],  p.  31.  The 
true  altitude  h  is  derived  from  the  reading  of  the  vertical  circle 
by  applying  the  index  correction  with  proper  sign  and  then 
subtracting  the  refraction  correction  (Table  I).  The  polar 
distance  is  found  by  taking  from  the  Ephemeris  (Table  of 
Circumpolar  Stars)  the  apparent  declination  of  the  star  and 
subtracting  this  from  90°. 

*  The  image  of  a  star  would  be  practically  a  point  of  light  in  a  perfect  telescope, 
but,  owing  to  the  imperfections  in  the  corrections  for  spherical  and  chromatic 
aberration,  the  image  is  irregular  in  shape  and  has  an  appreciable  width.  The 
image  of  the  star  should  be  bisected  with  the  horizontal  cross  hair. 


OBSERVATIONS  FOR  LATITUDE  105 

Example  i. 

Observed  altitude  of  Polaris  at  upper  culmination  =  43°  37'; 
index  correction  =  +30";  declination  =  +88°  44'  35". 

Vertical  circle  =  43°  37'  oo" 

Index  correction  = +30 

Observed  altitude  =43    37    30 
Refraction  correction    =  i    oo 

True  altitude  =43    36    30 

Polar  distance  =    i     15    25 

Latitude  =  42°  21'  05" 

Since  the  vertical  circle  reads  only  to  i'  the  resulting  value  for  the 
latitude  must  be  considered  as  reliable  only  to  the  nearest  i'. 

Example  2. 

Observed  altitude  of  51  Cephei  at  lower  culmination  =  39° 
33' 30";  index  correction  =  o";  declination  =  +-87°  n'  2$". 

Observed  altitude          =  39°  33'  30" 
Refraction  correction    =  i    09 


True  altitude  =39    32    21 

Polar  distance  =    2    48    35 

Latitude  =  42°  20'  56" 

66.   Latitude  by  Altitude  of  Sun  at  Noon. 

The  altitude  of  the  sun  at  noon  (meridian  passage)  may  be 
determined  by  placing  the  line  of  sight  of  the  transit  in  the  plane 
of  the  meridian  and  observing  the  altitude  of  the  upper  or  lower 
limb  of  the  sun  when  it  is  on  the  vertical  cross  hair.  The  watch 
time  at  which  the  sun  will  pass  the  meridian  may  be  computed 
by  converting  i2h  local  apparent  time  into  Standard  or  local 
mean  time  (whichever  is  used)  as  shown  in  Arts.  28  and  35. 
Usually  the  direction  of  the  meridian  is  not  known,  so  the  maxi- 
mum altitude  of  the  sun  is  observed  and  assumed  to  be  the  same 
as  the  meridian  altitude.  On  account  of  the  sun's  changing 
declination  the  maximum  altitude  is  'not  quite  the  same  as  the 
meridian  altitude;  the  difference  is  quite  small,  however,  usually 
a  fraction  of  a  second,  and  may  be  entirely  neglected  for  obser- 
vations made  with  the  engineer's  transit  or  the  sextant.  The 
maximum  altitude  of  the  upper  or  lower  limb  is  found  by  trial, 


106  PRACTICAL  ASTRONOMY 

the  horizontal  cross  hair  being  kept  tangent  to  the  limb  as  long 
#s  it  continues  to  rise.  When  the  observed  limb  begins  to  drop 
below  the  cross  hair  the  altitude  is  read  from  the  vertical  arc 
and  the  index  correction  is  determined.  The  true  altitude  of 
the  centre  of  the  sun  is  then  found  by  applying  the  corrections  for 
index  error,  refraction,  semidiameter,  and  parallax.  In  order 
to  compute  the  latitude  it  is  necessary  to  know  the  sun's  declina- 
tion at  the  instant  the  altitude  was  taken.  If  the  longitude  of 
the  place  is  known  the  sun's  declination  may  be  corrected  as 
follows:  The  west  longitude  of  the  place  is  the  same  as  the 
Greenwich  Apparent  Time,  because  the  Local  Apparent  Time 
of  the  observation  is  oh.  The  Greenwich  Mean  Time  is  found 
from  the  Greenwich  Apparent  Time  by  adding  or  subtracting 
the  equation  of  time.  The  declination  for  G.  M.  N.  must  be 
increased  or  decreased  by  an  amount  equal  to  the  "  variation 
per  hour  "  multiplied  by  the  number  of  hours  in  the  Greenwich 
Mean  Time.  In  case  the  time  of  the  observation  is  noted  on  a 
timepiece  keeping  Greenwich  Mean  Time  or  Standard  Time,  it 
is  not  necessary  to  employ  the  longitude  because  the  number 
of  hours  since  Greenwich  Mean  Noon  is  at  once  known.  An 
error  of  im  in  the  time  will  never  cause  an  error  greater  than  i" 
in  the  computed  declination.  The  latitude  is  found  by  apply- 
ing equation  [2],  p.  30. 

Example  i.  Observed  maximum  altitude  of  the  sun's  lower  limb,  Feb.  16, 
1916  =  34°  46'  30";  index  correction  =  +  i';  the  longitude  is  71°  04'  30"  west; 
the  declination  of  the  sun  at  Greenwich  Mean  Noon  is  S  12°  41'  2o".g;  the  varia- 
tion per  hour  =  -|-  51  ".49;  the  equation  of  time  is  —  14™  i8s;  the  sun's  semi- 
diameter  =  1 6'  13". 

Observed  altitude       =  34°  46'.  5  Loc.  App.  Time  =  ooh  oom  oo8 

Index  correction         =        +  i.o  Longitude  =  4    44     18 

34    47  -5  Gr.  App.  Time  =  4*  44™  i8s 

Refraction  correction  =           i  .4  Equa.  time  =  14     18 

34  46  .  i  Gr.  Mean  Time  =  4A  5&m  36* 
Semidiameter             16  .2 

35  02  .3 
Parallax                      .£ 

Altitude  of  centre  35     02  .4              Decl.  at  G.  M.  N.  =  —  12°  41'  20". 9 

Declination  —  12    37  .1                  51" .49  X  4^.98  =            +4    16.  5 

Co-latitude  47°  39'. 5 

>    Latitude  42°   20'. 5 


OBSERVATIONS   FOR   LATITUDE  107 

Example  2. 

Observed  double  altitude  of  sun's  upper  limb  at  noon  on  Jan.  28,  1910  (with 
artificial  horizon),  =  59°  17' 40";  Eastern  Standard  time  =  11^57*"  ;  index  cor- 
rection =  +30";  declination  at  Greenwich  Mean  Noon  =  S  18°  21'  08";  hourly 
change  =  +39".  07;  semidiameter  =  16'  16". 

Decl.  at  G.  M.  N.  =  -  18°  21'  08"  Double  alt.         =  59 

39".  07X4*.  95  +3'  13"  I-C. 

Decl.  at  ii*  57™     =  -  18°  17'  55" 


29"  39'  OS- 
Refraction          =        —i    40 

29   37    25 
Semidiameter     =     — 16    16 


29    21    09 
Parallax  =  +8 


29     21      17 

Declination        —  18    17    55 

Co-latitude         =  47°  39'  12" 
Latitude  =  42°  20'  48" 

67.  By  the  Meridian  Altitude  of  a  Southern*  Star. 

The  latitude  may  be  found  from  the  observed  maximum  alti- 
tude of  a  star  which  culminates  south  of  the  zenith,  by  the 
method  of  the  preceding  article,  except  that  the  parallax  and 
semidiameter  corrections  become  zero,  and  that  it  is  not  necessary 
to  note  the  time  of  the  observation,  since  the  declination  of  the 
star  changes  so  slowly.  In  measuring  the  altitude  the  star's 
image  is  bisected  with  the  horizontal  cross  hair,  and  the  maxi- 
mum found  by  trial  as  when  observing  on  the  sun.  For  the 
method  of  finding  the  time  at  which  a  star  will  pass  the  me- 
ridian see  Art.  72. 

Example. 

Observed  meridian  altitude  of  6  Serpentis  =  51°  45';  index  correction  =  o; 
declination  of  star  =  +4°  05'  n". 

Observed  altitude  of  0  Serpentis           =       51°  45'  oo 
Refraction  correction  = —45 

5i    44    15 
Declination  of  star  =  +    4    05    1 1 

Co-latitude  =       47°  39'  04" 

Latitude  =       42°  20'  56" 

*  The  observer  is  assumed  to  be  in  the  northern  hemisphere. 


108  PRACTICAL  ASTRONOMY 

Constant  errors  in  the  measured  altitudes  may  be  eliminated 
by  combining  the  results  obtained  from  circumpolar  stars  with 
those  from  southern  stars.  An  error  which  makes  the  latitude 
too  great  in  one  case  will  make  it  too  small  by  the  same  amount 
in  the  other  case. 

68.   Altitudes  Near  the  Meridian. 

If  altitudes  of  the  sun  or  a  star  are  taken  near  the  meridian  they  may  be  reduced 
to  the  meridian  altitude  provided  the  latitude  and  the  times  are  known  approxi- 
mately. To  derive  the  formula  for  making  the  reduction  take  the  fundamental 
formula  given  in  Equa.  [8] 

sin  h  =  sin  L  sin  D  -\-  cos  L  cos  D  cos  P. 

This  may  be  transformed  into 

sin  h  =  cos  (L  —  D)  —  cos  L  cos  D  vers  P,  [70] 

or 

p 

sin  h  =  cos  (L  —  D}  —  cos  L  cos  D  X  2  sin2  —  [71] 

Transposing  and  denoting  by  hm  the  meridian  altitude,  90°  —  (L  —  Z)),  the  equa- 
tion becomes 

sin  hm  =  sin  h  +  cos  L  cos  D  vers  P,  [72] 

p 

or  sin  hm  =  sin  h  +  cos  L  cos  D   X  2  sin2  — .  [73} 

If  the  altitude  h  be  measured  and  the  corresponding  time  be  noted,  then  the  value 
of  P  becomes  known.  If  L  is  known  approximately,  then  the  second  term  may  be 
computed  and  hm,  or  90°  —  (L  —  D},  found  through  its  sine.  If  the  value  of  L 
derived  from  the  first  computation  does  not  agree  closely  with  the  assumed  value, 
a  second  computation  should  be  made  using  the  new  value  of  L.  When  observa- 
tions are  taken  within  a  few  minutes  of  the  meridian  (say  15™)  the  computation 
may  be  shortened  by  the  use  of  the  approximate  formula 

C"  =  112.5  X  Pz  X  cos  L  cos  D  sec  h  sin  i",  [74] 

in  which  C"  is  the  correction  in  seconds  of  arc  and  P  is  the  time  from  the  meridian 
expressed  in  seconds  of  time.  (Log  112.5  X  sin  i"  =  6.7367).  If  P  is  expressed  in 
minutes  the  formula  is 

C"  =  i ".9635  X  P2  X  cos  L  cos  D  sec  h.  [75] 

This  formula  may  be  derived  as  follows:  Transposing  Equa.  [73! 

P 


sin  hm  —  sin  h  =  2  cos  L  cos  D  sin2  —  •  [76] 


By  trigonometry 


p 

2  cos  |  (hm  +  h)  sin  %  (hm  -  h)  '=  2  cos  L  cos  D  sin2  —  •  [77] 


OBSERVATIONS   FOR  LATITUDE 


I09 


Since  h  is  nearly  equal  to  hm,  cos  \  (hm  +  h)  may  be  put  equal  to  cos  h\  placing 

C  =  hm  —  h,  the  equation  becomes 

p 
sin  |  C  =  cos  L  cos  D  sin2  —  sec  /t .  [78] 

C  and  P  are  both  small  angles  and  may  be  put  in  place  of  their  sines,  hence 

C  =  i  P2  X  cos  L  cos  D  sec  A.  [79] 

To  reduce  C  to  seconds  of  arc  and  P  to  seconds  of  time  the  left  member  must  be 
multiplied  by  sin  i"  and  the  right  by  (15  sin  i")2>  giving 

•C"  =  cos  L  cos  D  sec  h  X  P2  X  112.5  sin  i".  [74] 

In  using  this  formula  it  will  be  necessary  to  use  an  approximate  value  of  L;  a 
second  approximation  may  be  made  if  necessary. 

The  method  of  "  reduction  to  the  meridian  "  given  above  should  not  be  applied 
when  the  object  observed  is  far  from  the  meridian. 

Example  i. 

Observed  double  altitude  sun's  lower  limb  Jan.  28,  1910. 


Double  Alt.  0 

56°  44'    40" 
49     oo 
52     40 

56°  48'    47" 
I.  C.  +30" 


2)  56°  49'    17' 

28°  24'   38' 
Refr.  =        -i     46 


28      22       52 

s.d.  =      +16     16 


28    39     08 
par.  =  +8 


h.  =  28°  39'    16' 

log  cos  L  =  9.  86763 

log  cos  D  =  9.  97745 

log  vers  P  =  8. 17546 


8.  02054 

.  01048 

nat  sin  h  =    .  47953 


nat  sin  hm 

hm 

D 


.49001 
29°  20'    29' 
18    18     20 


Watch. 


iihi$ 


25* 

16  22 

17  10 


Watch  corr. 

E.  S.  T. 
App.  Noon 


+i 


igs 
19 


nhi7m  38s 
ii    57     21 


Hour  angle        =     39™  43s 

P=       9°  55'  45' 


Assumed  lat.  =      42°  30' 


L.  A.  N. 
Eq.T. 

L.  M.  T. 

Red.  for  long.  = 


1 2*  oom  oos 

+  13     03 

12    13     03 
15     42 


E.  S.  T.  =      ii    57  21 

Sun's  decl.  at 

G.  M.  N.     =-  i8°2i'  08" 

39".o7X4/l.3  =  2'  48" 

Cor'd  decl.       =  -  18°  18'  20" 


Co-lat.  =  47°  38'   49" 
Lat.  =  42°  21'    ii" 
A  recomputation,  using  the  corrected  latitude,  changes  this  result  to  42°  21'  04". 


HO  PRACTICAL  ASTRONOMY 

Example  2. 

Observed  altitude  of  y  Ceti  =  50°  33';  index  correction  =  —  i';  hour  angle  of 
7  Ceti  derived  from  observed  time  =  3W  i4s.2;  declination  =  -f-  2°  50'  30". 

log  cos  L  =  9.  8691 

log  cos  D  =9.  9995 

log  sec  h  =  o.  1967 

log  const  =  6.  7367 

2  log  P  =  4.  5765 

logC"       =1.3785 
C"  =  23".9 

Observed  altitude          =     50°  33'.  o 
Index  correction  =         —   i  .  o 


So    -32 
Refraction  correction    =         —  o 


True  altitude  =      50°  31'.  2 

Reduction  to  meridian  =          +0.4 

Meridian  altitude          =      50°  31'.  6 
Declination  =  -f-  2    50  .  5 

Co-latitude  =      47°  41'.  i 

Latitude  =      42°  18'.  9 

The  method  of  "ex-meridian  altitudes,"  as  it  is  sometimes  called,  may  be  used 
when  the  meridian  observation  is  lost  or  when  it  is  desired  to  increase  the  accuracy 
of  the  result  by  multiplying  the  number  of  observations. 

69.   Latitude  by  Altitude  of  Polaris  when  the  Time  is  Known. 

The  altitude  of  Polaris  varies  slowly  on  account  of  its  nearness  to  the  pole, 
hence,  if  the  sidereal  time  is  known,  the  latitude  may  be  found  accurately  by  an 
altitude  of  this  star  taken  at  any  time,  because  errors  in  the  time  have  a  rela- 
tively small  effect  upon  the  result.  Several  altitudes  should  be  taken  in  succession, 
and  the  time  noted  at  each  pointing  of  the  cross-hair  on  the  star.  For  obser- 
vations made  with  the  surveyor's  transit  and  covering  only  a  few  minutes'  time 
the  mean  of  the  altitudes  may  be  taken  as  corresponding  to  the  mean  of  the  observed 
times.  If  the  instrument  has  a  complete  vertical  circle,  half  of  the  observations 
should  be  made  with  the  instrument  in  the  reversed  position.  The  index  correc- 
tion should  be  determined  in  each  case.  In  order  to  compute  the  latitude  it  is 
necessary  to  know  the  hour  angle  of  the  star  at  the  instant  of  observation.  When 
a  common  watch  is  used  for  taking  the  time  the  star's  hour  angle  is  found  by  Equa. 
[47]  and  [37].  The  latitude  is  then  found  by  the  formula 

L  =  h  —  p  cos  P  -f-  \  sin  i'  p2  sin2  P  tan  h  [80] 

(log  \  sin  i'  =  6. 1627  —  10) 

The  derivation  of  the  formula  is  rather  complex  and  will  not  be  given  here.  It 
is  obtained  by  expanding  the  correction  to  h  in  a  series  in  ascending  powers  of 


OBSERVATIONS  FOR  LATITUDE 


III 


p,  the  small  terms  being  neglected.  The  sum  of  all  terms  after  that  containing 
pz  amounts  to  less  than  i"  and  these  have  therefore  been  omitted  in  Equa.  [80}. 
In  this  equation  p,  the  polar  distance,  is  expressed  in  minutes  of  arc.  Values  of 
the  last  term  may  be  taken  with  sufficient  accuracy  from  Table  VI.  The  alge- 
braic sign  of  the  second  term  is  deter- 
mined by  the  sign  of  cos  P;  the  third 
term  is  always  positive.  In  Fig.  54, 
P  is  the  pole,  S  the  star,  MS  the 
hour  angle,  and  PDA  the  almucantar 
through  P  or  circle  of  equal  altitudes. 
It  will  be  seen  that  the  term  p  cos  P 
is  approximately  the  distance  from  S 
to  E,  a  point  on  the  six-hour  circle 
PB;  the  distance  desired  is  SD,  the 
angular  distance  of  S  above  the  al- 
mucantar through  P.  The  last  term 
in  Equa.  [80]  is  approximately  equal 
to  DE,  each  term  in  the  series  giving 
a  closer  approximation  to  the  distance 
SD. 

Example.  FIG.  54 

Observed  Altitudes  of  Polaris,  Jan.  9,  1907. 


Index  correction  =  - 
to  be  13°  50.7-* 

tog* 

log  cos  P 


Watch. 
49m  26 
5i     45 
54 
56 


Altitudes. 
43°  28'.  s 
28.5 

14  28  .  o 

45  28  .  o 

!>  =  71'.  15;  P  is  found  from  the  observed  watch  times 


=  1.8522 
=  9-9872 


log  p  cos  P  =  1.8394 
p  cos  P  =  69'.  09 


Observed  alt. 
I.C. 


log  constant 
log/>2 
log  sin2  P 
log  tan  In 


Last  term 
43°    28'.  25 


=  6. 1627 
=  3- 7044 
=  8- 7578 
=  9.9762 

8.6011 
=  +  o'.  04 


Refraction 


ist.  and  2nd  terms 
Latitude 


43      27'.  25 

I  .01 


43 

i 


26'.  24 
09.05 


42°    17'.  19 


*  If  the  error  of  the  watch  is  known  the  sidereal  time  may  be  found  by  Equa.  [47]. 
For  method  of  finding  the  sidereal  time  by  observation  see  Chap.  XI.  The 
hour  angle  of  the  star  is  found  by  Equa.  [38],  p.  48. 


112  PRACTICAL  ASTRONOMY 

70.   Precise  Latitude  Determinations.  —  Talcott's  Method. 

The  most  precise  method  of  determining  latitude  is  that  known  as  "  Talcott's 
Method,"  which  requires  the  use  of  the  zenith  telescope.  In  making  observations 
the  observer  selects  two  stars,  one  north  of  the  zenith  and  one  south  of  it,  the  two 
zenith  distances  differing  by  only  a  few  minutes  of  angle,  and  the  right  ascensions 
differing  by  about  5  or  10  minutes  of  time.  For  the  best  results  the  zenith  dis- 
tances should  be  small  and  nearly  equal.  If  the  first  star  culminates  south  of 
the  zenith  the  telescope  is  turned  about  its  vertical  axis  until  the  stop  indicates 
that  it  is  in  the  meridian  and  on  the  south  side  of  the  zenith.  The  telescope  is 
tipped  until  the  sight  line  has  an  inclination  to  the  vertical  equal  to  the  mean  of 
the  two  zenith  distances.*  It  is  clamped  in  this  position  and  great  care  is  taken 
not  to  alter  its  inclination  until  the  observations  on  both  stars  are  completed. 
When  the  star  appears  in  the  field  the  micrometer  wire  is  set  so  as  to  bisect  the 
star's  image;  at  the  instant  of  culmination  the  setting  of  the  wire  is  perfected  and 
the  scale  of  the  spirit  level  is  read  at  the  same  time.  The  chronometer  (regulated 
to  local  sidereal  time)  should  be  read  when  the  bisection  is  made,  so  that  the  read- 
ing of  the  micrometer  may  be  corrected  if  the  star  was  not  exactly  on  the  meridian 
at  that  instant.  The  micrometer  screw  is  then  read.  The  telescope  is  then  turned 
to  the  north  side  of  the  meridian,  the  inclination  remaining  unchanged,  and  a 
similar  observation  made  on  the  other  star.  When  both  sets  of  micrometer  read- 
ings and  level  readings  have  been  obtained,  the  latitude  is  found  by  the  formula 

L  =  l(Da  +  Dn)  +  l  (ms  -  mn)  X  R  +  *  (/.  4  In)  +  I  (r«-rn),          [81] 

in  which  ms,  mn  are  the  micrometer  readings,  R  the  value  of  i  division  of  the  mi- 
crometer, ls,  In  the  level  corrections  (positive  when  the  north  reading  is  the  larger) 
and  rs,  rn  the  refraction  corrections.  Another  correction  must  be  added  in  case 
the  observation  is  taken  when  the  star  is  off  the  meridian. 

In  order  to  determine  latitude  by  this  method  with  the  precision  required  in 
geodetic  operations,  observations  are  made  on  several  nights,  and  on  each  night 
a  large  number  of  pairs  of  stars  is  observed.  By  this  method  a  latitude  may  be 
determined  within  about  o".  05  which  is  equivalent  to  nearly  5  feet  in  distance 
on  the  earth's  surface. 

Questions  and  Problems 

1.  Observed  maximum  altitude  sun's  lower  limb,  April  27,  1916  =  61°  16'. 
Index  correction  =  +  30".    The  approximate  Eastern  Standard  Time  is  IIA  42"* 
A.M.     The  sun's  declination  for  G.  M.  N.  is  N.  13°  49'  06". 6;  the  diff.  for  ih  = 
+  47".8i;  the  semidiameter  =  15'  55".    Compute  the  latitude  (Northern  Hemi- 
sphere). 

2.  The  observed  meridian  altitude  of  5  Crateris  =  33°  24';  index  correction  = 
+  30";  declination  of  star  =  —  14°  if  37".     Compute  the  north  latitude. 

3.  Observed  altitude  of  a  Cell  at  3A  o8m  49*  L.  S.  T.  =  51°  21';  I.  C.  =  —  i' 

*  In  order  to  compute  these  zenith  distances  it  is  necessary  to  know  a  rough 
value  of  the  latitude,  say  within  i'  or  2'.  This  may  be  found  by  an  observation 
with  the  zenith  telescope  using  one  of  the  preceding  methods. 


OBSERVATIONS  FOR  LATITUDE  113 

the  right  ascension  of  a  Ceti  =  2h  57™  24*.  8;  declination  =  +  3°  43'  22".     Com- 
pute the  latitude. 

4.  Observed  Altitude  of  Polaris,  41°  41'  30";  chronometer  time,  gh  44™  38*.  5 
(Loc.  Sid.  Time);  chronometer  correction,  —  34*.     The  R.  A.  of  Polaris  is  ih  25"* 
42s;  the  declination  is  +  88°  49'  29".     Compute  the  latitude. 

5.  Show  by  a  sketch  the  positions  of  the  following  three  points;  i.   Polaris 
at  greatest  elongation;  2.   Polaris  on  the  six-hour  circle;  3.    Polaris  at  the  same 
altitude  as  the  pole.     (See  Art.  69,  p.  no,  and  Fig.  28,  p.  36.) 

6.  What  is  the  most  favorable  position  of  the  sun  for  a  latitude  observation? 

7.  What  is  the  most  favorable  position  of  Polaris  for  a  latitude  observation? 

8.  Draw  a  sketch  showing  why  the  sun's  maximum  altitude  is  not  the  same  as 
the  meridian  altitude. 


CHAPTER  XI 
OBSERVATIONS   FOR   DETERMINING  THE   TIME 

71.  Observation  for  Local  Time. 

Observations  for  determining  the  local  time  at  any  place  at 
any  instant  usually  consist  in  finding  the  error  of  a  timepiece 
on  the  kind  of  time  which  it  is  supposed  to  keep.  To  find  the 
solar  time  it  is  necessary  to  determine  the  hour  angle  of  the  sun's 
centre.  To  find  the  sidereal  time  the  hour  angle  of  the  vernal 
equinox  must  be  measured.  In  some  cases  these  quantities 
cannot  be  measured  directly,  so  it  is  often  necessary  to  measure 
other  coordinates  and  to  calculate  the  desired  hour  angle  from 
these  measurements.  The  chronometer  correction  or  watch 
correction  is  the  amount  to  be  added  algebraically  to  the  read- 
ing of  the  timepiece  to  give  the  true  time  at  the  instant.  It  is 
positive  when  the  chronometer  is  slow,  negative  when  it  is  fast. 
The  rate  is  the  amount  the  timepiece  gains  or  loses  per  day; 
it  is  positive  when  it  is  losing,  negative  when  it  is  gaining. 

72.  Time  by  Transit  of  a  Star. 

The  most  direct  and  simple  means  of  determining  time  is  by 
observing  transits  of  stars  across  the  meridian.  If  the  line  of 
sight  of  a  transit  be  placed  so  as  to  revolve  in  the  plane  of  the 
meridian,  and  the  instant  observed  when  some  known  star 
passes  the  vertical  cross  hair,  then  the  local  sidereal  time  at  this 
instant  is  the  same  as  the  right  ascension  of  the  star  given  in 
the  Nautical  Almanac  for  the  date.  The  difference  between 
the  observed  chronometer  time  /  and  the  right  ascension  R 
is  the  chronometer  correction  T, 

or  T=R-t.  [82] 

If  the  chronometer  keeps  mean  solar  time  it  is  only  necessary 
to  convert  the  true  sidereal  time  R  into  mean  solar  time  by 

114 


OBSERVATIONS  FOR  DETERMINING   THE  TIME  115 

Equa.  [49],  and  the  difference  between  the  observed  and  com- 
puted times  is  the  chronometer  correction. 

The  transit  should  be  set  up  and  the  vertical  cross  hair  sighted 
on  a  meridian  mark  previously  established.  If  the  instrument 
is  in  adjustment  the  sight  line  will  then  swing  in  the  plane  .of 
the  meridian.  It  is  important  that  the  horizontal  axis  should 
be  accurately  levelled;  the  plate  level  which  is  parallel  to  this 
axis  should  be  adjusted  and  centred  carefully r  or  else  a  striding 
level  should  be  used.  Any  errors  in  the  adjustment  will  be 
eliminated  if  the  instrument  is  used  in  both  the  direct  and  re- 
versed positions,  provided  the  altitudes  of  the  stars  observed 
in  the  two  positions  are  equal.  It  is  usually  possible  to  select 
stars  whose  altitudes  are  so  nearly  equal  that  the  elimination 
of  errors  will  be  nearly  complete. 

In  order  to  find  the  star  which  is  to  be  observed,  its  approxi- 
mate altitude  should  be  computed  beforehand  and  set  off  on 
the  vertical  arc.  (See  Equa.  [i].)  In  making  this  computation 
the  refraction  correction  may  be  omitted,  since  it  is  not  usually 
necessary  to  know  the  altitude  closer  than  5  or  10  minutes. 
It  is  also  convenient  to  know  beforehand  the  approximate  time 
at  which  the  star  will  culminate,  in  order  to  be  prepared  for  the 
observation.  If  the  approximate  error  of  the  watch  is  already 
known,  then  the  watch  time  of  transit  may  be  computed  (Equa. 
[49])  and  the  appearance  of  the  star  in  the  field  looked  for  a 
little  in  advance  of  this  time.  If  the  data  from  the  Nautical 
Almanac  are  not  at  hand  the  computation  may  be  made,  with 
sufficient  accuracy  for  finding  the  star,  by  the  following  method : 
Compute  the  sun's  R.  A.  by  multiplying  4™  by  the  number  of 
days  since  March  22.  Take  the  star's  R.  A.  from  any  list  of 
stars  or  a  star  map.  The  star's  R.  A.  minus  the  sun's  R.  A. 
(Equa.  [49])  will  be  the  mean  local  time  within  perhaps  2m  or 
3W.  This  may  be  reduced  to  Standard  Time  by  the  method 
explained  in  Art.  35.  In  the  surveyor's  transit  the  field  of  view 
is  usually  about  i°,  so  the  star  will  be  seen  about  2™  before  it 
reaches  the  vertical  cross  hair.  Near  culmination  the  star's 


Il6  PRACTICAL  ASTRONOMY 

path  is  so  nearly  horizontal  that  it  will  appear  to  coincide  with 
the  horizontal  cross  hair  from  one  side  of  the  field  to  the  other. 
When  the  star  passes  the  vertical  cross  hair  the  time  should  be 
noted  as  accurately  as  possible.  A  stop  watch  will  sometimes 
be  found  convenient  in  field  obversations  with  the  surveyor's 
transit.  When  a  chronometer  is  used  the  "  eye  and  ear  method  " 
is  the  best.  (See  Art.  58.)  If  it  is  desired  to  determine  the 
latitude  from  this  same  star,  the  observer  has  only  to  set 
the  horizontal  cross  hair  on  the  star  immediately  after  making 
the  time  observation,  and  the  reading  of  the  vertical  arc  will 
give  the  star's  apparent  altitude  at  culmination.  (See  Art.  67.) 

The  computation  of  the  watch  correction  consists  in  finding 
the  true  time  at  which  the  star  should  transit  and  comparing 
it  with  the  observed  watch  time.  If  a  sidereal  watch  or  chro- 
nometer is  used  the  star's  right  ascension  is  at  once  the  local 
sidereal  time.  If  mean  time  is  desired,  the  true  sidereal  time 
must  be  converted  into  local  mean  solar  time,  or  into  Standard 
Time,  whichever  is  desired. 

Example. 

Observed  transit  of  a  Hydra  Sh  48™  5 8s.  5,  Eastern  time,  in 
longitude  5^  2om  west;  date  April  5,  1902.  From  the  almanac, 
the  star's  R.  A.  =9^  22™  48*4,  and  the  sun's  R.  A.  at  G.  M.  N.  = 
o*  51™  24S.6.  To  reduce  this  to  the  R.  A.  at  local  mean  noon 
take  from  Table  III  the  correction  for  5^  20™  which  is  +52S.6. 
The  corrected  R.  A.  =0^52  if.  2.  The  local  sidereal  time, 
which  is  9^  22™  48  .4,  is  then  reduced  to  Standard  Time  as 
follows  o 


R.  A.  Star 

=  gh 

22m 

48' 

?-4 

R.  A.  Sun 

=o 

52 

17 

.  2 

8 

30 

3i 

.  2 

c 

= 

I 

23 

.6 

Mean  Local  Time 

=  8 

2Q 

07 

.6 

Red.  to  75°  merid. 

= 

20 

oo 

.  0 

Eastern  Time 

=  8 

49 

07 

.6 

Watch  time 

=  8 

48 

58 

•  5 

Watch  slow  =  9s.  i 


OBSERVATIONS   FOR   DETERMINING   THE   TIME  117 

Transit  observations  for  the  determination  of  time  can  be 
much  more  accurately  made  in  low  than  in  high  latitudes. 
Near  the  pole  the  conditions  are  very  unfavorable. 

73.    Observations  with  Astronomical  Transit. 

The  method  just  described  is  in  principle  the  one  in  most  common  use  for  deter- 
mining sidereal  time  with  the  large  astronomical  transit.  Since  the  precision  at- 
tainable with  the  latter  instrument  is  much  greater  than  with  the  engineer's  transit, 
the  method  must  be  correspondingly  more  refined.  The  number  of  observations 
on  each  star  is  increased  by  using  a  large  number  of  vertical  threads,  commonly 
eleven.  These  times  of  transit  are  recorded  by  electric  signals  on  the  chrono- 
graph (see  Art.  56,  p.  93),  and  are  scaled  from  the  chronograph  sheet  to  hundredths 
of  a  second.  In  this  class  of  work  many  errors  which  have  been  assumed  to  be 
negligible  in  the  preceding  method  are  important  and  must  be  carefully  determined 
and  allowed  for.  The  instrument  has  to  be  set  into  the  plane  of  the  meridian  by 
means  of  repeated  trials,  and  there  is  always  a  small  remaining  error  in  the  azimuth 
of  the  sight  line.  This  error  in  azimuth  a  is  measured  by  comparing  the  observed 
times  of  rapidly  moving  (southern)  stars  and  slowly  moving  (circumpolar)  stars. 
The  correction  to  any  observed  time  for  the  effect  of  azimuth  error  is 

a  cos  h  sec  D.  [66] 

The  inclination  of  the  axis  to  the  horizon  b  is  measured  with  the  spirit  level  and 
the  observed  times  are  reduced  to  the  meridian  by  adding  the  correction 

b  sin  h  sec  D.  [67] 

The  error  in  the  sight  line  c  is  found  by  reversing  the  telescope  in  its  supports 
and  comparing  observations  made  in  the  two  positions.  The  correction  to  any 
observation  is 

c  sec  D.  [68] 

Corrections  are  also  made  for  the  effect  of  diurnal  aberration  and  sometimes  other 
minor  corrections. 

74.   Selecting  Stars  for  Transit  Observations. 

Before  the  observations  are  begun  the  observer  should  pre- 
pare a  list  of  stars  suitable  for  transit  observations.  This 
list  should  include  the  name  or  number  of  the  star,  its  magni- 
tude, the  approximate  time  of  culmination,  and  its  meridian 
altitude  or  its  zenith  distance.  The  right  ascensions  of  consec- 
utive stars  in  the  list  should  differ  by  sufficient  intervals  to  give 
the  observer  time  to  make  and  record  an  observation  and  pre- 
pare for  the  next  one.  The  stars  used  for  determining  time 
should  be  those  which  have  a  rapid  diurnal  motion,  that  is, 


Il8  PRACTICAL  ASTRONOMY 

stars  near  the  equator;  slowly  moving  stars  are  not  suitable 
for  time  determinations.  Very  faint  stars  should  not  be  selected 
unless  the  telescope  is  of  high  power  and  good  definition;  those 
smaller  than  the  fifth  magnitude  are  rather  difficult  to  observe 
with  a  small  transit,  especially  as  it  is  difficult  to  reduce  the 
amount  of  light  used  for  illuminating  the  field  of  view.  The 
selection  of  stars  will  also  be  governed  somewhat  by  a  consider- 
ation of  the  effect  of  the  different  instrumental  errors.  An  in- 
spection of  Table  B,  p.  88 ,  will  show  that  for  stars  near  the 
zenith  the  azimuth  error  is  zero,  while  the  inclination  error  is 
a  maximum;  for  stars  near  the  horizon  the  azimuth  error  is  a 
maximum  and  the  inclination  error  is  zero.  If  the  azimuth  of 
the  instrument  is  uncertain  and  the  inclination  can  be  accurately 
determined,  then  stars  having  high  altitudes  should  be  preferred. 
On  the  other  hand,  if  the  level  parallel  to  the  axis  is  not  a  sensi- 
tive one  and  is  in  poor  adjustment,  and  if  the  sight  line  can  be 
placed  accurately  in  the  meridian,  which  is  usually  the  case 
with  a  surveyor's  transit,  then  low  stars  will  give  the  more  accu- 
rate results.  With  the  surveyor's  transit  the  choice  of  stars  is 
somewhat  limited,  however,  because  it  is  not  practicable  to 
sight  the  telescope  at  much  greater  altitudes  than  about  70° 
with  the  use  of  the  prismatic  eyepiece  and  55°  or  60°  without 
this  attachment. 

Following  is  a  sample  list  of  stars  selected  for  observations 
in  a  place  whose  latitude  is  40°  N.,  longitude  77°  W.,  date  May  5, 
1910,  hours  between  Sh  and  gh  Eastern  time;  the  limiting  alti- 
tudes chosen  are  10°  and  65°.  The  right  ascension  of  the  mean 
sun  for  the  date  is  2h  50"*.  Adding  this  to  Sh  -  oSm  =  ^  52™, 
the  local  mean  time,  the  resulting  sidereal  time  is  ioh  42™, 
which  is  approximately  the  right  ascension  of  a  point  on  the 
meridian  at  Sh  E.  S.  T.  The  limiting  right  ascensions  are  there- 
fore ioh  42™  and  n*  42 m.  The  co-latitude  is  50°,  which  gives, 
for  altitudes  10°  and  65°,  the  limiting  declinations  +15°  and 
—  40°.  In  the  table  of  mean  places  for  1910  the  following 
stars  are  given: 


OBSERVATIONS  FOR  DETERMINING  THE   TIME 
MEAN   PLACES  FOR  1910 


Star. 

Magn. 

Rt.  Asc. 

Decl. 

/  Lconis 

57 

IO^    44TO  ^2S 

+  11°    Ol' 

d2  Chameleontis          .  . 

4   7 

IO      44      <7 

—  80     04 

46  Leonis  Minoris       

7     Q 

10     48     17 

+  34     42 

Groombridge  1  706  

6.7, 

10      tJ2      47 

+  78     iq 

a    Urs(B  Majoris  

2.O 

10     58     ii 

+  62     14 

ri    Octantis  •  

6.1 

IO       <O       5j8 

—  84     07 

p*  Lconis 

6    2 

II        O2        IO 

+     2       27 

il/    Ufscc  Afdjofis 

7    2 

II        O4        7,7 

+  44      cjo 

d    Lconis                        

2     7 

II       OO       IQ 

+  21      Ol 

v    UrscB  M&jofis     

7  .  7 

II         13         7,7 

+  ^      ^ 

5      CYdtBYlS      .              

7.0 

II      14      50 

—  14      17 

T    Leonis  
X    Draconis  

5-1 
4.0 

II    23    19 

II      26     04 

+   3     21 
+  69      50 

£  HydrcB 

7     8 

II       28       74. 

—  TT        22 

u  Lconis 

4  4 

II       32       20 

—     O       2O 

X.  UYSCR  MajoTis            .  .        .... 

2,     O 

ii     41     18 

+  48       17 

/3  Lconis  

2  .  2 

ii      44     28 

+  iq     o=; 

From  this  list  there  are  found  seven  stars  whose  declinations 
and  right  ascensions  fall  within  or  very  close  to  the  required 
limits.  In  the  following  list  the  times  of  transit  and  the  alti- 
tudes have  been  computed  roughly  but  with  sufficient  accuracy 
to  identify  the  stars. 

OBSERVING   LIST   FOR   TRANSIT   OBSERVATIONS 


Star. 

Magn. 

Approx.  E.S.T. 

Approx.  Alt. 

/  Lconis    

r  .  7 

8/1  oom 

61°   01' 

pp  Leonis  

6.2 

8     18 

<2       27 

8    Crateris 

3Q 

8       7.0 

3C        47. 

T    Lconis 

c    J 

ow 

8        70 

57.       21 

£    Hvdrce 

3-8 

8     44 

O       •*  A 

18     7,8 

v  Lconis      

4.  4 

8     48 

40      4O 

/3  Leonis  

2  .  2 

O       OO 

6<     o(? 

75.   Time  by  Transit  of  the  Sun. 

The   apparent  solar   time  may    be   directly  determined  by 
observing  the  watch  times  when  the  west  and  east  limbs  of  the 


120  PRACTICAL  ASTRONOMY 

.sun  cross  the  meridian.  The  mean  of  the  two  readings  is  the 
watch  time  for  the  instant  of  Local  Apparent  Noon  or  i2hM. 
apparent  time.  This  apparent  time  is  to  be  converted  into 
mean  time  and  then  into  Standard  Time.  If  only  one  limb 
•of  the  sun  can  be  observed  the  time  of  transit  of  the  centre  may 
be  found  by  adding  or  subtracting  the  "  time  of  semidiameter 
passing  the  meridian,"  which  is  given  in  the  Nautical  Almanac. 

Example. 

Observed  transit  of  sun  on  Jan.  28,  1910,  longitude  4*  44™  i8s  W.     Time  of 
transit  of  W.  limb  =  nh  54™  53*;  E.  limb  =  nh  57™  ns;  mean  of  two  limbs  = 
ii*  56™  02s.  o. 
L.  A.  T.         =  12*    oom  oos  Equa.  of  T.  at  G.  M.  N.     =  13™  oo8.  60 

Equa.  T.        =       +13      03  -o  o8. 485  X  4h  .93  = 2_._39 

L.  M.  T.        =  12      13     03  .o  Cor'd.  Equa.  T.  =  13™  02*.  99 

Red.  to  75°    =       —  15     42  .o 
E.  S.  T.          =11      57      21  .  o 

Watch  time  =  11 56     02  .  o 

Watch  slow  =  im    19*.  o 

76.   Time  by  an  Altitude  of  the  Sun. 

The  apparent  solar  time  may  be  determined  by  measuring 
the  altitude  of  the  sun  when  it  is  not  near  the  meridian,  and 
then  solving  the  PZS  triangle  for  the  angle  at  the  pole,  which  is 
the  hour  angle  of  the  sun  east  or  west  of  the  meridian.  The 
west  hour  angle  of  the  sun  is  the  local  apparent  time.  The 
observation  is  made  by  measuring  several  altitudes  in  quick 
succession  and  noting  the  corresponding  instants  of  time.  The 
mean  of  the  observed  altitudes  is  assumed  to  correspond  to  the 
mean  of  the  observed  times,  that  is,  the  curvature  of  the  path 
of  the  sun  is  neglected.  The  error  caused  by  neglecting  the 
correction  for  curvature  is  very  small  provided  the  sun  is  not 
near  the  meridian  and  the  series  of  observations  extends  over 
but  a  few  minutes'  time,  say  iom.  The  measurement  of  alti- 
tude must  of  course  be  made  to  the  upper  or  the  lower  limb 
and  a  correction  applied  for  the  semidiameter.  The  observa- 
tions may  be  made  in  two  sets,  half  the  altitudes  being  taken 
on  the  upper  limb  and  half  on  the  lower  limb,  in  which  case  no 
semidiameter  correction  is  required.  The  telescope  should  be 


OBSERVATIONS  FOR  DETERMINING  THE   TIME  121 

reversed  between  the  two  sets  if  the  instrument  has  a  complete 
vertical  circle.  The  mean  of  the  altitudes  must  be  corrected 
for  index  error,  refraction,  and  parallax,  and  for  semidiametet 
if  but  one  limb  is  observed.  The  declination  must  be  corrected 
by  adding  to  the  declination  at  G.  M.  N.  the  hourly  change  mul- 
tiplied by  the  number  of  hours  since  G.  M.  N.  It  is  necessary 
for  this  purpose  to  know  the  approximate  Greenwich  Mean 
Time.  If  the  watch  used  is  keeping  Standard  Time  the  G.  M.  T. 
is  found  at  once.  (Art.  35.)  If  the  watch  is  not  more  than 
2m  or  2™  in  error  the  effect  on  the  computed  declination  will  be 
negligible  for  observations  made  with  small  instruments.  If  the 
longitude  is  known  the  declination  may  be  corrected  by  first 
computing  an  approximate  value  of  the  local  time  and  adding 
this  to  the  longitude,  obtaining  the  approximate  G.  M.  T. 
With  this  approximate  G.  M.  T.  the  declination  may  be  cor- 
rected and  the  whole  computation  repeated.  It  will  seldom  be 
necessary  to  make  a  third  computation.  In  order  to  compute 
the  hour  angle  the  latitude  of  the  place  must  be  known.  The 
hour  angle  of  the  sun's  centre  P  is  then  found  by  means  of  one 
of  the  formulae  of  Art.  19.*  When  the  value  of  P  is  found  it 
is  converted  into  hours,  minutes  and  seconds,  and  if  the  sun  is 
east  of  the  meridian  it  is  subtracted  from  i2h  to  obtain  the  local 
(civil)  apparent  time;  if  astronomical  time  is  desired  it  should 
be  subtracted  from  24^.  This  apparent  time  is  then  converted 
into  mean  time  by.  adding  or  subtracting  the  equation  of  time. 

*  If  tables  of  log  versed  sines,  in  addition  to  the  usual  tables,  are  available, 
then  the  following  formula  will  be  found  convenient: 

[83] 


cos  L  cos  D 
In  case  P  is  greater  than  90°  the  formula  below  may  be  substituted: 

versp,  =  sin  A  +  0030  +  1^ 
cos  L  cos  D 

where  P'  =  180°  -  P. 

The  sum  or  difference  in  the  numerator  must  be  computed  with  natural  func- 
tions and  the  remainder  of  the  computation  performed  by  logarithms. 


122  PRACTICAL  ASTRONOMY 

The  equation  of  time  must  be  corrected  for  the  time  elapsed 
since  G.  M.  N.  The  resulting  mean  time  is  to  be  converted 
into  Standard  Time,  to  which  the  watch  is  regulated.  The 
difference  between  the  computed  result  and  the  mean  of  the 
observed  watch  readings  is  the  watch  correction. 

The  most  favorable  conditions  for  an  accurate  determination 
of  time  by  this  method  are  when  the  sun  is  on  the  prime  vertical 
and  when  the  observer  is  at  the  equator.  When  the  sun  is 
east  or  west  it  is  rising  or  falling  at  its  most  rapid  rate  and  an 
error  of  i'  in  the  altitude  produces  less  error  in  the  calculated 
hour  angle  than  does  i'  error  when  the  sun  is  near  the  meridian. 
The  nearer  the  observer  is  to  the  equator  the  greater  is  the  in- 
clination of  the  sun's  path  to  the  horizon,  and  consequently  the 
greater  its  rise  or  fall  per  second  of  time.  If  the  observer  were 
at  the  equator  and  the  declination  zero,  the  sun  would  rise  or 
fall  i'  in  4s  of  time.  In  the  example  given  below  the  rise  is 
i'  in  about  8s  of  time.  When  the  observer  is  near  the  pole 
the  method  is  practically  useless.  Observations  on  the  sun 
when  it  is  very  close  to  the  horizon  should  be  avoided,  however, 
even  when  the  sun  is  near  the  prime  vertical,  because  the  errors 
in  the  tabulated  refraction  correction  due  to  variations  in  the 
temperature  and  pressure  of  the  air  are  likely  to  be  large.  Ob- 
servations should  not  be  made  when  the  altitude  is  less  than 
about  10°  if  this  can  readily  be  avoided. 

Example. 

Observation  of  Sun's  Altitude  for  Time,  Nov.  28,  1905.  Lat.  42°  21'  N.  Long. 
7i°  04.  5'  W. 

Altitude  Watch  (Eastern  Time) 

Lower  limb  (   14°    41'  8h   39™    42*  A.M. 

Tel.  dir.  \   15°    oo'  8     42       19 

Upper  limb  (   15°    55'  8     45       34 

Tel.  rev.  1   16°    08'  8     47       34 

Mean  =15°    26'.  o         Mean         =  &h   43™    47s.  2  A.M. 

Refraction  and  parallax    =  3  .3          G.  M.  T.   =  ih   43™    47s.  2  (approx.) 

£    =    I5°      22'.  7 


OBSERVATIONS   FOR   DETERMINING   THE   TIME  123 

L=    42°  2i'.o     sec   0.13133          Decl.  at  G.  M.  N.    =    -   21°  14'   54" 

h=    15    22.7                                    -26".8iXA  73     =  -46 
p  =  in    15.7      esc   0.03061 


Corrected  decl.     =  -    21°  15'  40' 

2s  =  168°  59'. 4                                                                 P=  in°   15'   4o' 

s  =    84°  29'.  7     cos   8.  98196 
s  —  h  =    69°  07'.  o     sin,  9.  97049 

Eq.  t.     =  i2m    ©4s.  29 

2)9.11439                 .846X1.73     =  '                1-46 


log  sin  \  P  =    9.  55719  I2m    02s.  83 

*  P  =  21°  08'   45" 
P   =42°   17'    30" 
=  a*  4TO  ios.  o 


L.  A.  T.  =  gh    iom    50s.  o 

Eq.  t.  12       02  .8 

M.  L.  T.  =  8^    58™   47s .  2 

15       42  .o 


Eastern  time          =  8*    43™    05*  .  2 
Watch  time  =8      43       47-2 


Watch  fast  42s.  o 

77.   Time  by  the  Altitude  of  a  Star. 

The  method  of  the  preceding  article  may  be  applied  equally 
well  to  an  observation  on  a  star.  In  this  case  the  parallax  and 
semidiameter  corrections  are  zero.  If  the  star  is  west  of  the  meri- 
dian the  computed  hour  angle  is  the  star's  true  hour  angle; 
if  the  star  is  east  of  the  meridian  the  computed  hour  angle  must 
be  subtracted  from  24^.  The  sidereal  time  is  then  found  by 
adding  the  right  ascension  of  the  star  to  its  hour  angle.  If 
mean  time  is  desired  the  sidereal  time  thus  found  is  to  be  con- 
verted into  mean  solar  time  by  Art.  34.  Since  it  is  easy  to  select 
stars  in  almost  any  position  it  is  desirable  to  eliminate  errors  in 
the  measured  altitudes  by  taking  two  observations,  one  on  a 
star  which  is  nearly  due  east,  the  other  on  one  about  due  west. 
The  mean  of  these  two  results  will  be  nearly  free  from  instru- 
mental errors,  and  also  from  errors  in  the  assumed  value  of  the 
observer's  latitude.  If  a  planet  is  used  it  will  be  necessary  to 
know  the  G.  M.  T.  with  sufficient  accuracy  for  correcting  the 
right  ascension  and  declination. 


124 


PRACTICAL  ASTRONOMY 


Example. 

Observed  altitude  of  Jupiter  \ 
7i°  17'.  5 

Observed  altitude  =  44°  55' 
Index  correction  =  —  i 
Refraction  correction  =  —  i 


h  =  44°    53'- o 


p  = 


42°  iS'.o 
44  53  .  o 
66  41  .4 


2  s  =  152 
s  =    76 


112'.  4 
56  .  2 


R.  A.  at  G.  M.  N. 
Hourly  change 


12*.  53  X  - 
Corrected  R.  A. 


395 


6h  ig 


-17*.  5 
6h  i8w    59s. 


),  Jan.  9,  1907.     Lat. 
Eastern  time 

Decl.  at  G.  M.  N.  = 
Hourly  change        = 
G.  M.  T. 

12^.53  X  i".oo     = 
Corrected  decl.       = 

=  42°  iS'.o;     Long.  = 

=   7h  32™  02s 

+  23°     18'      22".  0 
+     l".00 

I2A  32™  O2S  (approx.) 

+  12".  5 
+  23°  18'    34".  5 

P  = 

66°  41'    25" 

•5 

s  -  L  =  34°  38'.  2 

esc    o.  24537 

s  —  h   =  32     03  .  2 

sin    9.  72486 

s  —  p    =  10     14  .  8 

sec   o.  00698 

5       =76       56  .  2 

cos   9.35416 

2)9.33i37 

3                    log  tan  \ 

•  P  =  9.  66568 

395                                i 

P=   24°   50' 

57" 

5 
8 

P  =   49°   4i' 

54" 

=  2/1      jgW 

47s.  6 

=    20^  4iw 

I  2s.  4 

R. 

A.  =     6h   1  8™ 

59s.  8 

Sid.  Time  =   27^  oom    12s.  2 

The  local  sidereal  time  is  therefore  3^  oow  1 2s .  2  when  the  watch  reading  is 
7 A  32™  O2S.  The  error  of  the  watch  may  be  found  by  reducing  the  sidereal  time 
to  Eastern  Time.f 

78.  Time  by  Transit  of  Star  over  Vertical  Circle  through  Polaris.  J 
In  making  observations  by  this  method  the  line  of  sight  of  the  telescope  is  set 
in-  the  vertical  plane  through  Polaris  at  any  (observed)  instant  of  time,  and  the 
time  of  transit  of  some  southern  star  across  this  plane  is  observed  immediately 
afterward;  the  correction  for  reducing  the  star's  right  ascension  to  the  true  sidereal 
time  of  the  observation  is  then  computed  and  added  to  the  right  ascension.  The 
advantages  of  the  method  are  that  the  direction  of  the  meridian  does  not  have  to 
be  established  before  time  observations  can  be  begun,  and  that  the  interval  which 
must  elapse  between  the  two  observed  times  is  so  small  that  errors  due  to  the 
instability  of  the  instrument  are  reduced  to  a  minimum. 

The  method  of  making  the  observation  is  as  follows:  Set  up  the  instrument  and 
level  carefully;  sight  the  vertical  cross  hair  on  Polaris  (and  clamp)  and  note  and 
record  the  watch  reading;  then  revolve  the  telescope  about  the  horizontal  axis, 

*  The  parallax  correction  is  negligible  for  the  planet  Jupiter;  it  should  not  be 
neglected,  however,  in  case  of  the  inner  planets. 

t  See  Problem  8,  page  138,  for  data  for  computing  the  Eastern  Standard  Time 
from  this  observation. 

t  For  a  complete  discussion  of  this  method  see  a  paper  by  Professor  George 
O.  James,  in  the  Jour.  Assoc.  Eng.  Soc.,  Vol.  XXXVII,  No.  2;  also  Popular  As- 
tronomy, No.  172.  A  method  applicable  to  larger  instruments  is  given  by  Professor 
Frederick  H.  Sears,  in  Bulletin  No.  5,  Laws  Observatory,  University  of  Missouri. 


OBSERVATIONS  FOR  DETERMINING  THE  TIME 


125 


being  careful  not  to  disturb  its  azimuth;  set  off  on  the  vertical  arc  the  altitude 
of  some  southern  star  (called  the  time-star)  which  will  transit  about  4™  or  5W 
later;  note  the  instant  when  this  star  passes  the  vertical  cross  hair.  It  will  be  of 
assistance  in  making  the  calculations  if  the  altitude  of  each  star  is  measured 
immediately  after  the  time  has  been  observed.  The  altitude  of  the  time-star 
at  the  instant  of  observation  will  be  so  nearly  equal  to  its  meridian  altitude  that 
no  special  computation  is  necessary  beyond  what  is  required  for  ordinary  transit 
observations.  If  the  times  of  meridian  transit  are  calculated  beforehand  the 
actual  times  of  transit  may  be  estimated  with  sufficient  accuracy  by  noting  the 
position  of  Polaris  with  respect  to  the  meridian.  If  Polaris  is  near  its  elongation 
then  the  azimuth  of  the  sight 
line  will  be  a  maximum.  In 
latitude  40°  the  azimuth  of 
Polaris  for  1910  is  about  i°  32'; 
a  star  on  the  equator  would 
then  pass  the  vertical  cross  hair 
nearly  4™  later  than  the  com- 
puted time  if  Polaris  is  at  east- 
ern elongation  (see  .Table  B, 
p.  88) .  If  Polaris  is  near  west- 
ern elongation  the  star  will  tran- 
sit earlier  by  this  amount.  In 
order  to  eliminate  errors  in  the 
adjustment  of  the  instrument, 
observations  should  be  made 
in  the  erect  and  inverted  posi- 
tions of  the  telescope  and  the 
two  results  combined.  A  new 
setting  should  be  made  on 
Polaris  just  before  each  obser- 
vation on  a  time-star. 

In  order  to  deduce  an  expression  for  the  difference  in  time  between  the  meridian 
transit  and  the  observed  transit  let  R  and  R0  be  the  right  ascensions  of  the  stars, 
6"  and  50  the  sidereal  times  of  transit  over  the  cross  hair,  P  and  P0  the  hour  angles 
of  the  stars,  the  subscripts  referring  to  Polaris.  Then  by  Equa.  [37],  p.  48, 

P  =S  -  R 

and  P0  =  S0  -  RQ-, 

subtracting,  P0  -  P  =  (R  -  RQ)  -  (S  -  So).  [85] 

The  quantity  S  —  S0  is  the  observed  interval  of  time  between  the  two  observa- 
tions expressed  in  sidereal  units.  If  an  ordinary  watch  is  used  the  interval  must 
be  reduced  to  sidereal  units  (Table  III).  Equa.  [85]  may  then  be  written 

Po  -  P  =  (R  -  R0)  -  (T  -  To)  -  C,  [86] 

where  T  and  T0  are  the  actual  watch  readings  and  C  is  the  correction  to  reduce 
this  interval  to  sidereal  time. 

In  Fig.  55  let  P0  be  the  position  of  Polaris  when  it  is  observed;  P,  the  celestial 


126  PRACTICAL  ASTRONOMY 

north  pole;  Z,  the  zenith  of  the  observer;  and  S,  the  time-star  in  the  position  in 
which  it  is  observed.  Notice  that  when  S  is  passing  the  cross-hair,  Polaris  is  not 
in  the  position  P0,  but  has  moved  westward  (about  P)  by  an  angle  equal  to  the 
(sidereal)  time  interval  between  the  two  observations.  Let  p0  be  the  polar  distance 
of  Polaris;  z  and  z0,  the  zenith-distances  of  the  two  stars;  and  h  and  ho  their 
altitudes. 

Then  in  the  triangle  PoPS, 

sin  5       _    sin/>0 
sin  P0PS  ~  sin  P^S ' 

or  sin  S  =  sin  PoPS  sin  p0  sec  (z  +  z0) 

=  sin  (Po  —  P)  sin  p0  cosec  (h  +  h0),  [87] 

by  equa.  [85]. 
In  triangle  PZS, 

sin  ( —  P)  _  sin  z 
sin  5       ~~  cos  L1 

or  sin  (-  P)  =  sin  5  cos  h  sec  L.  [88] 

Substituting  the  value  of  sin  S  in  equa.  [87]. 

sin  ( -  P)  =  sin  p0  sin  (P0  -  P)  cosec  (A  +  A0)  cos  /*  sec  L.  [89] 

Since  P  and  p0  are  small  the  angles  may  be  substituted  for  their  sines,  and 

-  P  =  po  sin  (Po  -  P)  cosec  (h  +  h0)  cos  h  sec  L.  [go] 

If  the  altitudes  h  and  &o  have  not  been  measured  the  factor  cos  h  may  be  replaced 
by  sin  (L  —  D)  and  cosec  (h  +  hQ]  may  be  replaced  by  sec  (D  —  c)  with  an  error 
of  only  a  few  hundredths  of  a  second,  where  D  is  the  declination  of  the  time-star, 
and  c  is  the  correction  given  in  Table  I  at  the  end  of  the  Nautical  Almanac. 

In  this  method  the  latitude,  L,  is  supposed  to  be  known.  If  it  is  not  known, 
then  the  altitudes  of  the  stars  must  be  measured  and  L  computed.  It  will  usually 
be  accurate  enough  to  assume  that  the  observed  altitude  of  the  time-star  is  the 
same  as  the  meridian  altitude,  and  apply  equa.  (2);  otherwise  a  correction  may  be 
made  by  formula  (74)  or  (75).  The  latitude  may  also  be  found  from  the  altitude 
of  the  polestar,  using  the  method  of  Art.  69. 

After  the  value  of  P  (in  seconds  of  time  *)  has  been  computed  it  is  added  to  the 
right  ascension  of  the  time-star  to  obtain  the  local  sidereal  time  of  the  observation 
on  this  star.  This  sidereal  time  may  then  be  reduced  to  mean  local  and  then  to 
standard  time  and  the  watch  correction  obtained. 

If  it  is  desired  to  find  the  azimuth  of  the  line  of  sight  this  may  be  done  by 
computing  a  by  the  formula 

a  =  P  sec  h  cos  D.  [91] 

*  The  factor  4  has  been  introduced  in  the  following  example  in  order  to  reduce 
minutes  of  angle  to  seconds  of  time. 


OBSERVATIONS    FOR   DETERMINING   THE  TIME  127 

The  above  method  is  applicable  to  transit  observations  made  with  a  small 
instrument.  For  the  large  astronomical  transit  a  more  refined  method  of  making 
the  reductions  must  be  used. 

Example. 

Observation  of  o  Virginis  over  Vertical  Circle  through  Polaris;  Lat.,  42°  21'  N., 
Long.,  4h  44m  *%8-3  W.;  Date,  May  8,  1906. 

Observed  time  on  Polaris  =  &h    35™    58* 

Observed  transit  of  o  Virginis         =8      39       43 

Diff.       =  3™    45* 

oom     268.3 

24       35  .4  L  =       42°     21' 

D  =  +    9       IS 


R  -  RQ  =     IOA    35™    50s. 
T  -  T0  =  3       45  -o  L  -  D  =       33°    06' 

C  =  .6 

Po      7i'.85 

Po  —  P  =     IOA     32™     05^3         log  £0  =  1-8564 

=  158°     oi'.3  log  sin  (Po  — P)  =  9-5732 

log  sec  (D  —c)    =  0.0044  D  =  +  9°i5' 

log  sin  (L  -  D)   =  9.7373  c  =  +  i°o6'.s 

log  sec  L  =0.1313 


log  4  =  0.6021    D  —  c  =      8°o8'.s 

logP 


P  =  -8os.3o 


The  true  sidereal  time  may  now  be  found  by  subtracting  im  20*.  3  from  the  right 
ascension  of  o  Virginis.  The  complete  computation  of  the  watch  correction  is  as 
follows: 

R  =  i2h   oo™    26s.  3 

P  =        -   i       20  .  3 


s  = 
*.- 

c  = 

M.L.T.= 

Eastern  time       = 
Watch  time        = 

Watch  fast          = 

ii* 

3 

59TO 
02 

o6s. 
23  • 

0 

6 

8* 

56W 

i 

42S. 

27  . 

4 
9 

8" 

55W 
i5 

i4s. 
41  . 

5 

7 

& 
8 

39m 

39 

32S. 

43 

8 

10s. 

a 

79.   Time  by  Equal  Altitudes  of  a  Star. 

If  the  altitude  of  a  star  is  observed  when  it  is  east  of  the  meridian  at  a  certain 
altitude,  and  the  same  altitude  of  the  same  star  again  observed  when  the  star  is 
west  of  the  meridian,  then  the  mean  of  the  two  observed  times  is  the  watch  reading 


128  PRACTICAL  ASTRONOMY 

for  the  instant  of  transit  of  the  star.  It  is  not  necessary  to  know  the  actual 
value  of  the  altitude  employed,  but  it  is  essential  that  the  two  altitudes  should  be 
equal.  The  disadvantage  of  the  method  is  that  the  interval  between  the  two 
observations  is  inconveniently  long. 

80.   Time  by  Two  Stars  at  Equal  Altitudes. 

In  this  method  the  sidereal  time  is  determined  by  observing  when  two  stars 
have  equal  altitudes,  one  star  being  east  of  the  meridian  and  the  other  west.  If 
the  two  stars  have  the  same  declination  then  the  mean  of  the  two  right  ascensions 
is  the  sidereal  time  at  the  instant  the  two  stars  have  the  same  altitude.  As  it  is 
not  practicable  to  find  pairs  of  stars  having  exactly  the  same  declination  it  is  neces- 
sary to  choose  pairs  whose  declinations  differ  as  little  as  possible  and  to  introduce 
a  correction  for  the  effect  of  this  difference  upon  the  sidereal  time.  It  is  not 
possible  to  observe  both  stars  directly  with  a  transit  at  the  instant  when  their 
altitudes  are  equal;  it  is  necessary,  therefore,  to  first  observe  one  star  at  a  certain 
altitude  and  to  note  the  time,  and  then  to  observe  the  other  star  at  the  same  alti- 
tude and  again  note  the  time.  The  advantage  of  this  method  is  that  the  actual 
value  of  the  altitude  is  not  used  in  the  computations;  any  errors  in  the  altitude 
due  either  to  lack  of  adjustment  of  the  transit  or  to  abnormal  refraction  are  there- 
fore eliminated  from  the  result,  provided  the  two  altitudes  are  made  equal.  In 
preparing  to  make  the  observations  it  is  well  to  compute  beforehand  the  approx- 
imate time  of  equal  altitudes  and  to  observe  the  first  star  two  or  three  minutes 
before  the  computed  time.  In  this  way  the  interval  between  the  observations 
may  be  kept  conveniently  small.  It  is  immaterial  whether  the  east  star  is  observed 
first  or  the  west  star  first,  provided  the  proper  change  is  made  in  the  computation. 
If  one  star  is  faint  it  is  well  to  observe  the  bright  one  first;  the  faint  star  may  then 
be  more  easily  found  by  knowing  the  time  at  which  it  should  pass  the  horizontal 
cross  hair.  The  interval  by  which  the  second  observation  follows  the  time  of 
equal  altitudes  is  nearly  the  same  as  the  interval  between  the  first  observation 
and  the  time  of  equal  altitudes.  It  is  evident  that  in  the  application  of  this 
method  the  observer  must  be  able  to  identify  the  stars  he  is  to  observe.  A  star 
map  is  of  great  assistance  in  making  these  observations. 

The  observation  is  made  by  setting  the  horizontal  cross  hair  a  little  above  the 
easterly  star  2m  or  3™  before  the  time  of  equal  altitudes,  and  noting  the  instant 
when  the  star  passes  the  horizontal  cross  hair.  Before  the  star  crosses  the  hair 
the  clamp  to  the  horizontal  axis  should  be  set  firmly,  and  the  plate  bubble  which 
is  perpendicular  to  the  horizontal  axis  should  be  centred.  When  the  first  obser- 
vation has  been  made  and  recorded  the  telescope  is  then  turned  toward  the  westerly 
star,  care  being  taken  not  to  alter  the  inclination  of  the  telescope,  and  the  time 
when  the  star  passes  the  horizontal  cross  hair  is  observed  and  recorded.  It  is 
well  to  note  the  altitude,  but  this  is  not  ordinarily  used  in  making  the  reduction. 
If  the  time  of  equal  altitudes  is  not  known,  then  both  stars  should  be  bright  ones 
that  are  easily  found  in  the  telescope.  The  observer  may  measure  an  approxi- 
mate altitude  of  first  one  and  then  the  other,  until  they  are  at  so  nearly  the  same 
altitude  that  both  can  be  brought  into  the  field  without  changing  the  inclination 
of  the  telescope.  The  altitude  of  the  east  star  may  then  be  observed  at  once  and 


OBSERVATIONS   FOR   DETERMINING   THE   TIME 


129 


the  observation  on  the  west  star  will  follow  by  only  a  few  minutes.  If  it  is  desired 
to  observe  the  west  star  first,  it  must  be  observed  at  an  altitude  which  is  greater 
than  when  the  east  star  is  observed  first.  In  this  case  the  cross  hair  is  set  a  little 
below  the  star. 

In  Fig.  56  let  nesw  represent  the  horizon,  Z  the  zenith,  P  the  pole,  Se  the  easterly 
star,  and  Sw  the  westerly  star. 
Let  Pe  and  Pw  be  the  hour 
angle  of  Se  and  Sw,  and  let 
HSeSw  be  an  almucantar,  or 
circle  of  equal  altitudes. 

From  Equa.  [37],  for  the 
two  stars  Se  and  Sw,  the 
sidereal  time  is 

S  =  RW  4~  PW 

S  =  Re-  Pe* 

Taking  the  mean  value  of  S, 


-Pe 


_,_ RW~\~RC  . 

o  —  I 


from  which  it  is  seen  that 
the  true  sidereal  time  equals 
the  mean  right  ascension 
corrected  by  half  the  differ- 
ence in  the  hour  angles.  To 
derive  the  equation  for  cor- 
recting the  mean  right  ascension  so  as  to  obtain  the  true  sidereal  time  let  the 
fundamental  equation 

sin  h  =  sin  D  sin  L  •+  cos  D  cos  L  cos  P  [8] 

be  differentiated  regarding  D  and  P  as  the  only  variables,  then  there  results 


o  =  sin  L  cos  D  —  cos  D  cos  L  sin  P  -r=-  —  cos  L  cos  P  sin  D, 

dD 


from  which  may  be  obtained 


d_P_ 
dD 


tan  L       tan  D 


[93] 


[94] 


sin  P       tan  P 

If  the  difference  in  the  declination  is  small,  dD  may  be  replaced  by  f  (Dw—  De), 
in  which  case  dP  will  be  the  resulting  change  in  the  hour  angle,  or  |  (Pw  —Pe]. 
The  equation  for  the  sidereal  time  then  becomes 

"tan  L      tan 


Rw+Re  ,    Dw  -  De 

~T~ 


[95] 


in  which  (Dw  —  De)  must  be  expressed  in  seconds  of  time.      D  may  be  taken 
as  the  mean  of  De  and  Dw.    The  value  of  P  would  be  the  mean  of  Pe  and  Pw  if 


*  Pe  is  here  taken  as  the  actual  value  of  the  hour  angle  east  of  the  meridian. 


130 


PRACTICAL  ASTRONOMY 


the  two  stars  were  observed  at  the  same  instant,  but  since  there  is  an  appreciable 
interval  between  the  two  times  P  must  be  found  by 


, 


2  2 

If  the  west  star  is  observed  first,  then  the  last  term  becomes  a  negative  quantity. 
Strictly  speaking  this  last  term  should  be  converted  into  sidereal  units,  but  the 
effect  upon  the  result  is  usually  very  small.  In  regard  to  the  sign  of  the  correction 
to  the  mean  right  ascension  it  should  be  observed  that  if  the  west  star  has  the 
greater  declination  the  time  of  equal  altitudes  is  later  than  that  indicated  by  the 
mean  right  ascension.  In  selecting  stars  for  the  observation  the  members  of  a 
pair  should  differ  in  right  ascension  by  6  to  8  hours,  or  more,  according  to  the 
declinations.  Stars  above  the  equator  should  have  a  longer  interval  between 
them  than  those  below  the  equator.  On  account  of  the  approximations  made  in 
deriving  the  formula  the  declinations  should  differ  as  little  as  possible.  If  the 
declinations  do  not  differ  by  more  than  about  5°,  however,  the  result  will  usually 
be  close  enough  for  observations  made  with  the  engineer's  transit.  From  the 
extensive  star  list  now  given  in  the  Nautical  Almanac  it  is  not  difficult  to  select 
a  sufficient  number  of  pairs  at  any  time  for  making  an  accurate  determination 
of  the  local  time.  Following  is  a  short  list  taken  from  the  American  Ephemeris 
and  arranged  for  making  an  observation  on  April  30,  1912. 

LIST  FOR  OBSERVING  BY  EQUAL  ALTITUDES 
Lat.,  42°  21'  N.     Long.,  4h  44™  iSs  W.     Date,  Apr.  30,  1912. 


Stars. 

Magn. 

Sidereal  time 
of  equal  alti- 
tudes. 

Kastern  time 
of  equal  alti- 
tudes. 

Observed 
times. 

a  Corona  Borealis  
ft  Tauri 

2-3 

1.8 

10*   28™ 

jh   3gw 

a  Boolis  ... 

O.  2 

£"  Geminorum      '       

4 

.10     37 

7      47 

a  Bo'dtis  .  .          

O.2 

5  Gevninoruw  

•2  .  C 

10     48 

7     58 

p  Bo'otis    .        

*.6 

a*  Geminorum  
•jf  Hydras       

1-9 
•2.  C 

II       00 

8     10 

P  Argus  
ft  Herculis  

2.9 
2.8 
•2    r 

II        10 

II     19 

8       20 

8     29 

a.  Serpentis  
a.  Canis  Minoris  
ft  Herculis' 

2.7 

0-5 
2.8 

ii     35 

8     45 

8  Gevninoruw               .... 

•?  .  e 

ii     5i 

9     01 

a  Serpentis                   .... 

2  .  7 

3  C&ncri                   .    ... 

3.8 

12       02 

9     12 

a  Serpenlis              

2  .  7 

e  Hydras                  

7  .  C 

12        II 

9     21 

ft  Libra                

2.9 

a  HydrcB  -               

2  .  I 

12       20 

9     3° 

l3  Herculis  

2.8 

y  Cancri  

4.9 

12       32 

9     42 

OBSERVATIONS   FOR   DETERMINING   THE   TIME  131 

Following  is  an  example  of  an  observation  for  time  by  the  method  of  equal  alti- 
tudes. 

Example. 

Lat,  42°  21'  N.     Long.,  4*  44™  18*  W.     Date,  Dec.  14,  1905. 

Star.  Rt.  Asc.  Decl.  Watch. 

a  Ceti  (E)  2h     57™    22*.  i  +  3°    43'    69".  i  5*  i8m    oos 

d  Aquilte  (W)      19      20      43-6  +  2      55     44  .  o  522       13 

Mean 
Diff. 


2) 

P= 

=  57°    36'      31".  5 

Mean  R.  A.     =  23^   09™     02s.  8 
Corr.  —  01       41  .  o 


23* 

7 

36 

4 

02s. 

13  . 

8 
5 
7 

2)~ 

i 

19' 
48' 

56" 
25" 

.6 
.  I 

5&    20m 

04 

06*.  5 

Dw  — 

a 

nfis 

1  2".  6 

RA 

>  7h 

4om 

52*. 

, 

2 

Sid.  Time        =   23^  07™  2is.8         D-D 

tfs  =17  30  43  .  2  log^—  e  =  i.  9861  (n)                  i.  9861  (n) 

"  &  *6m  ?8S  6  loS  tan  L       =  9-  9598  log  tan  D  =  8.  7650 

C'  =    '  55  .'  2  !og  esc  P       =o.  0735  log  cot  P  =  9.  8024 


M.  L.  T.          -    5*   35-     43'.  4  =  ^  (n)  .  £  S6535   (n) 

15       42  .o 


Eastern  time  =    5^    2om     ois.  4 

Watch  time     =5      20      06  . 5    Corr-  "  IO1  •  ° 

Watch  fast      =  $«.  i 

8 1 .  Formula  [94]  may  be  made  practically  exact  by  means  of  the  following  device. 
Applying  Equa.  [8]  to  each  star  separately  and  subtracting  one  result  from  the 
other  we  obtain  the  equation* 

.  „       tan  L  tan  AD       tan  D  tan  AZ)      tan  D  tan  AZ>  .    . 

smAP=    -&^~        tanf        -ssrr--  ""^  [97] 

where  AZ)  is  half  the  difference  in  the  declinations  and  AP  is  the  correction  to 
the  mean  right  ascension.  If  sin  AP  and  tan  AD  are  replaced  by  their  arcs 
and  the  third  term  dropped,  this  reduces  to  Equa.  [94],  except  that  AZ>  and  AP 
are  finite  differences  instead  of  infinitesimals.  In  order  to  compensate  for  the 
errors  thus  produced  let  AD  be  increased  by  a  quantity  equal  to  the  difference 
between  the  arc  and  the  tangent  (Table  C);  and  let  a  correction  be  added  to  the 
sum  of  the  first  two  terms  to  allow  for  the  difference  between  the  arc  and  sine  of 
AP  (Table  C).  With  the  approximate  value  of  AP  thus  obtained  the  third 

*  Chauvenet,  Spherical  and  Practical  Astronomy,  Vol.  I,  p.  199. 


PRACTICAL  ASTRONOMY 


term  of  the  series  may  be  taken  from  Table  D.  By  this  means  the  precision  of 
the  computed  result  may  be  increased,  and  the  limits  of  AZ)  may  therefore  be 
extended  without  increasing  the  errors  arising  from  the  approximations. 


TABLE  C.     CORRECTIONS  TO  BE  ADDED  TO  AZ>  AND  AP 
(Equa.  [97],  Art.  81) 


Arc  or  sine. 

Correction  to 
A£>. 

Correction  to 
AP. 

Arc  or  sine. 

Correction  to 
A£>. 

Correction  to 
AP. 

5 

100 

8 

o.oo 

s 

0.00 

s 

800 

s 
0.90 

s 
o-45 

200 

O.OI 

O.OI 

850 

1.  08 

o-S4 

300 

0.05 

0.02 

900 

1.29 

0.64 

400 

O.II 

O.06 

95° 

I-5I 

0.76 

500 

O.22 

O.II 

IOOO 

1.77 

0.88 

600 

0.38 

0.19 

1050 

2.05 

1.02 

650 

0.48 

0.24 

1  100 

2-35 

I.I7 

700 

O.6o 

0.30 

1150 

2.69 

1-34 

75° 

0.74 

o-37 

I2OO 

3.06 

i-52 

TABLE  D.     CORRECTION  TO  BE  ADDED  TO  AP  * 

(Equa.  [97],  Art.  81) 


AP  (in  seconds  of  time). 

2d 
term. 

IOOS 

200S 

300s 

400s 

5oos 

6oo« 

7oos 

800* 

9oos 

1000s 

9 

8 

S 

s 

s 

s 

s 

5 

8 

8 

s 

100 

O.OO 

O.OI 

O.O2 

0.04 

0.07 

0.  10 

0.13 

0.17 

0.21 

0.26 

2OO 

O.OI 

O.O2 

O.O5 

O.o8 

0.13 

0.19 

0.26 

o-34 

0.43 

o-53 

300 

O.OI 

0.03 

0.07 

0.13 

0.2O 

0.29 

o-39 

0.51 

0.64 

0.79 

4OO 

O.OI 

0.04 

O.IO 

0.17 

0.26 

0.38 

0.52 

0.68 

0.86 

i.  06 

500 

O.OI 

O.05 

O.I2 

0.21 

o-33 

0.48 

0.65 

0.85 

1.07 

1.32 

600 

O.O2 

O.O6 

O.I4 

0.25 

0.40 

0.57 

0.78 

1.02 

1.28 

i-59 

700 

O.02 

0.07 

0.17 

0.30 

0.46 

0.67 

0.91 

1.18 

1.50 

1.85 

800 

O.02 

0.08 

0.19 

0-34 

0-53 

0.76 

1.04 

i-35 

1.71 

2.  II 

900 

O.O2 

0.10 

0.21 

0.38 

o-59 

0.86 

1.17 

1.52 

i-93 

2.38 

IOOO 

0.03 

O.II 

0.24 

O.42 

0.66 

°-95 

1.30 

1.69 

2.14 

2.64 

IIOO 

0.03 

0.12 

0.26 

0.47 

o-73 

1.05 

1.42 

1.86 

2.36 

2.9I 

I20O 

0.03 

0.13 

0.29 

0.51 

0.79 

1.14 

i-55 

2.03 

2-57 

3-17 

*  The  algebraic  sign  of  this  term  is  always  opposite  to  that  of  the  second  term. 


OBSERVATIONS   FOR   DETERMINING  THE   TIME  133 

Example. 

Compute  the  time  of  equal  altitudes  of  a  Bootis  and  e,  Geminorum  on  Jan.  i, 
1912,  in  latitude  42°  21'.  R.  A.  a  Bootis  =  i4h  nw  37s-  9§;  decl.  =  +  19° 
38'  15".  2.  R.  A.  L  Geminorum  =  jh  20™  i6s.  85;  decl.  =  +  27°  58'  30".  8. 

I4h  Ixm    37s.  98  27°  58'   30".  8 

7    20       16  .85  19     38     15  -2 

2)    6h  51"*  2is.  i?  2)8 


4OS.  5')  AZ)  =  4°   10'    07".  8 

P  =  51°  25'      08".  4  =  iooos.  52 

Corr.,  Table  C   =        1.7? 

AZ)  =ioo2s.  29 

log  AZ>         =  3.  000993  log   AD  =  3.  00099 

log  tan  L     =9.  959769  log  tan  D  =  9.  64462 

log  esc  P     =o.  106945  log  cot  P  =  9.90187 

3.067707  2.54748 

ist  term               =  n68*.7i  ad  term  =    —  352-7<5 
2d  term                =~352  -?6 

A.P  (approx.)  =  8is*.95 
Corr.,  Table  C  =  -f  .  48 
Corr.,  Table  D  =  +  .63 


AP  =  +   817*.  06 

=  +     13™  37s- 06 
Mean  R.  A.  =  IO    45      57 .  42 


Sid.  Time  of  Equal  Alt.   =   \vh   59™  34*.  48 

For  refined  observations  the  inclination  of  the  vertical  axis  should  be  measured 
with  a  spirit  level  and  a  correction  applied  to  the  observed  time.  With  the  engi- 
neer's transit  the  only  practicable  way  of  doing  this  is  by  means  of  the  plate-level 
which  is  parallel  to  the  plane  of  motion  of  the  telescope.  If  both  ends  of  this 
level  are  read  at  each  observation,  O  denoting  the  reading  of  the  object  end  and  E 
the  eye  end  of  the  bubble,  then  the  change  in  the  inclination  is  expressed  by 


where  d  is  the  angular  value  of  one  scale  division  in  seconds  of  arc.    The  correction 
to  the  mean  watch  reading  is 


30  sin  S  cos  D  30  cos  L  sin  Z 
in  which  5  may  be  taken  from  the  Azimuth*  tables  or  Z  may  be  found  from  the 
measured  horizontal  angle  between  the  stars.  If  the  west  star  is  observed  at  a 
higher  altitude  than  the  east  star  (bubble  nearer  objective),  the  correction  must 
be  added  to  the  mean  watch  reading.  If  it  is  applied  to  the  mean  of  the  right 
ascensions  the  algebraic  sign  must  be  reversed. 

*  See  Arts.  82  and  109  for  the  method  of  using  these  tables. 


134 


PRACTICAL  ASTRONOMY 


82.  The  correction  to  the  mean  right  ascension  of  the  two  stars  may  be  con- 
veniently found  by  the  following  method,  provided  the  calculation  of  the  paral- 
lactic  angle,  S  in  the  PZS  triangle,  can  be  avoided  by  the  use  of  tables.  Publica- 
tion No.  1 20  of  the  U.  S.  Hydrographic  Office  gives  value  of  the  azimuth  angle 
for  every  whole  degree  of  latitude  and  declination  and  for  every  iom  of  hour  angle. 
The  parallactic  angle  may  be  obtained  from  these  tables  (by  interpolation)  by 
interchanging  the  latitude  and  the  declination,  that  is,  by  looking  up  the  declin- 
ation at  the  head  of  the  page  and  the  latitude  in  the  line  marked  "  Declination." 
For  latitudes  under  23°  it  will  be  necessary  to  use  Publication  No.  71. 

In  taking  out  the  angle  the  table  should  be  entered  with  the  next  less  whole 
degree  of  latitude  and  of  declination  and  the  next  less  iow  of  hour  angle,  and  the 
corresponding  tabular  angle  written  down;  the  proportional  parts  for  minutes 
of  latitude,  of  declination,  and  of  hour  angle  are  then  taken  out  and  added  alge- 
braically to  the  first  angle.  The  result  may  be  made  more  accurate  by  working 


FIG.  57 

from  the  nearest  tabular  numbers  instead  of  the  next  less.  The  instructions  given 
in  Pub.  1 20  for  taking  out  the  angle  when  the  latitude  and  declination  are  of 
opposite  sign  should  be  modified  as  follows.  Enter  the  table  with  the  supplement 
of  the  hour  angle,  the  latitude  and  declination  being  interchanged  as  before,  and 
the  tabular  angle  is  the  value  of  51  sought. 

Suppose  that  two  stars  have  equal  declinations  and  that  at  a  certain  instant 
their  altitudes  are  equal,  A  being  east  of  the  meridian  and  B  west  of  the  meridian. 
If  the  declination  of  B  is  increased  so  that  the  star  occupies  the  position  C,  then 
the  star  must  increase  its  hour  angle  by  a  certain  amount  x  in  order  to  be  again 
on  the  almucantar  through  B.  Half  of  the  angle  x  is  the  desired  correction. 
In  Fig.  57  BC  is  the  increase  in  declination;  BD  is  the  almucantar  through 
A,  B  and  D;  and  CD  is  the  arc  of  the  parallel  of  declination  through  which  the 
star  must  move  in  order  to  reach  BD.  The  arcs  BD  and  CD  are  not  arcs  of 
great  circles,  and  the  triangle  BCD  is  not  strictly  a  spherical  triangle,  but  it  may 


OBSERVATIONS   FOR   DETERMINING   THE   TIME  135 

be  shown  that  the  error  is  usually  negligible  in  observations  made  with  the  engi- 
neer's transit  if  BCD  is  computed  as  a  spherical  triangle  or  even  as  a  plane  triangle. 
The  angle  ZBP  is  the  angle  S  and  DEC  is  90°  -  S.  The  length  of  the  arc  CD 
is  then  EC  cot  S,  or  (Dw  —  De)  cot  S.  The  angle  at  P  is  the  same  as  the  arc 
CD'  and  equals  CD  sec  D.  If  (Dw  —  De)  is  expressed  in  minutes  of  arc  and  the 
correction  is  to  be  in  seconds  of  time,  then,  remembering  that  the  correction  is 
half  the  angle  x, 

Correction  =  2  (Dw  —  De)  cot  S  sec  D.  [98] 

D  should  be  taken  as  the  mean  of  the  two  declinations,  and  the  hour  angle,  used 
in  finding  S,  is  half  the  difference  in  right  ascension  corrected  for  half  the  watch 
interval.  • 

The  trigonometric  formula  for  determining  the  correction  for  equal  altitudes  is 

tan  ~=  sin  ^  cot  }  (Si  +  S«)  sec  }  (A  +  A) ,  [99] 

By  substituting  arcs  for  the  sine  and  tangent  this  reduces  to  the  equation  given 
above,  except  that  the  mean  of  6*1  and  52  is  not  exactly  the  same  as  the  value  of  S 
obtained  by  using  the  mean  of  the  hour  angles. 

The  example  on  p.  131  worked  by  this  method  is  as  follows.  From  the  azimuth 
tables,  using  a  declination  of  42°,  latitude  3°,  and  hour  angle  3^  50™,  the  approxi- 
mate value  of  S  is  44°  05'.  Then  from  the  tabular  differences,  — 

Correction  for  21'  decl.  =  —  22' 
Correction  for  20'  lat.     =  +  07 
Correction  for  26*  h.  a.  =  +  02 
The  corrected  value  of  S  is  therefore  43°  52' 

2  (Dw  -  De)  =  -  96'.  84  log    =  i.  9861  (n) 
log  cot  S  =  0.0172 
log  sec  D  =  o.  0007 

log  corr.    =  2.  0040  (n) 
log  corr.   =  —  ioos.  9 

This  solution  is  sufficiently  accurate  for  observations  made  with  the  engineer's 
transit,  provided  the  difference  in  the  declinations  of  the  two  stars  is  not  greater 
than  about  5°  and  the  other  conditions  are  favorable.  For  larger  instruments 
and  for  refined  work  this  formula  is  not  sufficiently  exact. 

The  equal-altitude  method,  like  all  of  the  preceding  methods,  gives  more  precise 
results  in  low  than  in  high  latitudes. 

83.  Rating  a  Watch  by  Transit  of  a  Star  over  a  Range. 

If  the  time  of  transit  of  a  fixed*  star  across  some  well-defined 
range  can  be  observed,  the  rate  of  a  watch  may  be  quite  accu- 
rately determined  without  knowing  its  actual  error.  The 
disappearance  of  the  star  behind  a  building  or  other  object 

*  A  planet  should  not  be  used  for  this  observation. 


136  PRACTICAL  ASTRONOMY 

when  the  eye  is  placed  at  some  definite  point  will  serve  the  pur- 
pose. The  star  will  pass  the  range  at  the  same  instant  of  sidereal 
time  every  day.  If  the  watch  keeps  sidereal  time,  then  its 
reading  should  be  the  same  each  day  at  the  time  of  the  star's 
transit  over  the  range.  If  the  watch  keeps  mean  time  it  will 
lose  3 m  55s. 9 1  per  sidereal  day,  so  that  the  readings  on  successive 
days  will  be  less  by  this  amount.  If,  then,  the  passage  of  the 
star  be  observed  on  a  certain  night,  the  time  of  transit  on  any 
subsequent  night  is  computed  by  multiplying  3m55s.Qi  by  the 
number  of  days  intervening  and  subtracting  this  correction 
from  the  observed  time.  The  difference  between  the  observed 
and  computed  times  divided  by  the  number  of  days  is  the  daily 
gain  or  loss.  After  a  few  weeks  the  star  will  cross  the  range  in 
daylight,  and  it  will  be  necessary  before  this  occurs  to  transfer 
to  another  star  which  transits  later  in  the  same  evening.  In 
this  way  the  observations  may  be  carried  on  indefinitely. 

84.   Time  Service. 

The  Standard  Time  used  for  general  purposes  in  this  country 
is  determined  by  observations  at  Washington  and  is  sent  out 
to  all  parts  of  the  country  east  of  the  Rocky  Mountains  by 
means  of  electric  signals  transmitted  over  the  lines  of  the  tele- 
graph companies.  For  the  territory  west  of  the  Rocky  Moun- 
tains the  time  is  determined  at  the  Mare  Island  Navy  Yard 
and  distributed  by  telegraphic  signals.  The  error  of  the  sidereal 
clock  of  the  observatory  is  determined  at  frequent  intervals 
by  observing  star  transits.  The  sidereal  clock  is  then  compared 
with  a  mean-time  clock,  by  means  of  a  chronograph,  and  the 
error  of  this  clock  on  mean  time  is  computed.  The  mean- time 
clock  is  then  compared  with  another  mean-time  clock  especially 
designed  for  sending  the  automatic  signals.  When  the  error 
of  this  sending  clock  is  found  it  is  "  set  "  (to  Eastern  Standard 
Time)  by  accelerating  or  retarding  the  motion  of  the  pendulum 
until  the  error  is  reduced  to  a  negligible  quantity.  The  series 
of  signals  sent  out  each  day  begins  at  11^55™  A.M.,  Eastern 
time,  and  continues  for  five  minutes.  The  clock  mechanism 


OBSERVATIONS   FOR  DETERMINING  THE  TIME  137 

is  arranged  to  break  the  circuit  at  the  end  of  each  second; 
this  makes  a  click  on  every  telegraph  sounder  on  the  line,  or 
a  notch  on  the  sheet  of  a  chronograph  placed  in  the  circuit. 
The  end  of  each  minute  is  shown  by  the  omission  of  the  55th 
to  59th  seconds  inclusive,  except  for  the  noon  signal,  which 
is  preceded  by  a  ten-second  interval.  During  this  ten-second 
interval  the  local  circuits  controlling  the  time-balls,*  which 
are  dropped  by  this  same  signal,  are  thrown  into  the  main 
circuit.  The  signals  sent  out  in  this  way  are  seldom  in  error 
by  an  amount  greater  than  one  or  two  tenths  of  a  second.  The 
break  in  the  circuit  which  occurs  at  the  instant  of  noon,  Eastern 
time,  drops  all  the  time-balls,  corrects  the  clocks  placed  in  the 
circuit,  and  gives  a  click  on  every  telegraph  sounder  on  the  line. 
In  many  seaports  the  wireless  telegraph  lines  are  also  thrown 
into  the  circuit  and  the  signal  thus  made  available  at  sea. 

Questions  and  Problems 

1.  Compute  the  approximate  Eastern  time  of  transit  of  Regulus  over  the  me- 
ridian 71°  04/0  West  of  Greenwich  on  March  21,  1908.     The  R.  A.  of  Regulus 
is  ioh  03  m  2QS. i ;  Rs  at  G.  M.  N.  =  23^  54™  23*. 99.  . 

2.  Compute  the  error  of  the  watch  from  the  data  given  in  prob.  6,  p.  169. 

3.  Observed  time  of  transit  of  8  Capricorni  over  the  vertical  circle  through 
Polaris,    Oct.    26,    1906.     Latitude  =  42°  i8'-5;   longitude  =  4h  45™  07*.      Ob- 
served watch  time  of  transit  of  Polaris  =  7^  iom  20s;  of  8  Capricorni  =  jh  13™  28s, 
Eastern   Time.      Declination   of    Polaris  =  +  88°  48'  31  ".3;    right   ascension  = 
ih  26™  37s. 9.     Declination  of  8  Capricorni  =  —  16°  3.3'  02". 8;  right  ascension  = 
2Ih   4Im  538.3.       The    right    ascension   of    the    Mean    Sun    at    Local    Mean 
Noon  =  14^  i6m  34S.6.     Compute  the  error  of  the  watch  on  Eastern  Time. 

4.  Time  observation  on  May  3,  1907,  in  latitude  42°  2i'.o,  longitude  4h  44™ 
i8s.o.     Observed  transit  of  Polaris  =  7h  i6m  i7'.o;  of  /j.  Hydra  =  7^  i8OT  5o8.5. 
Decl.  of  Polaris  =  +  88°  48'  28"-3;  R.  A.  =  i*  24™  50-". 2.    Decl.  of  /*  Hydra  = 

-  16°  21'  53". 2;  R.  A.=  ioh  2im  36*.!.  R.  A.  of  Mean  Sun  at  G.  M.  N.  =  2h  40™ 
S6S.63.     Find  the  error  of  the  watch. 

5.  Observation  for  time  by  equal  altitudes,  Dec.  18,  1904. 

R.  A.  Decl.  Watch. 

a  Tauri  (E)  4h  3<>m    29s.oi  +  16°    18'    59^.9         7^  34™    56* 

a  Pegasi  (W)          22     59       61  .12  +  14      41     43    .7         7     39       45 

Lat.  =  42°  28'.o;  long.  =  4h  44™  15*  .o.     R.  A.  Mean  Sun  at  G.  M.  N.  =  17** 
46™  40s-38. 

*  Time-balls  are  now  in  use  in  the  principal  ports  on  the  Atlantic,  Pacific, 
and  Gulf  coasts  and  on  the  Great  Lakes. 


138  PRACTICAL  ASTRONOMY 

6.  Time  by  equal  altitudes,  Oct.  13, 1906. 

R.  A.  Decl.  Watch 

vOphiuchi  (W)  ijh  53™    523.15  -9°    45'   34"-6  7h  i3m    49* 

i  Ceti  (E)  o     14       40  .99  —  9      20     25  .7  7     28       25 

Lat.  =  42°  18';  long.  =  4h  45™  o6*.8.     R.  A.  of  Mean  Sun  at   G.  M.  N.  = 
13^  24^  32s.s6. 

7.  Show  by  differentiating  Equa.  [8]  that  the  most  favorable  position  of  the 
sun  for  a  time  observation  is  on  the  prime  vertical.     The  differential  coefficients 

'-fj-  and  —   should  be  a  minimum  to  give  the  greatest  accuracy.      The  expres- 
sions obtained  may  be  simplified  by  means  of  Equa.  [12]  and  [n]. 

8.  Compute  the  watch  correction  from  the  observation  given  on  p.  124.     The 
R.  A.  of  the  mean  sun  at  G.  M.  N.  on  Jan.  9,  1907,  was  19^  nm  298.49. 


CHAPTER  XII 
OBSERVATIONS  FOR  LONGITUDE 

85.  Method  of  Measuring  Longitude. 

The  measurement  of  the  difference  in  longitude  of  two  places 
depends  upon  a  comparison  of  the  local  times  of  the  places  at 
the  same  absolute  instant  of  time.  One  important  method 
is  that  in  which  the  timepiece  is  carried  from  one  station  to 
the  other  and  its  error  on  local  time  determined  in  each  place. 
The  most  precise  method,  however,  and  the  one  chiefly  used 
in  geodetic  work,  is  the  telegraphic  method,  in  which  the  local 
times  are  compared  by  means  of  electric  signals  sent  through  a 
telegraph  line.  Other  methods,  most  of  them  of  inferior  accu- 
racy, are  those  which  depend  upon  a  determination  of  the  moon's 
position  (moon  culminations,  eclipses,  occultations)  and  upon 
eclipses  of  Jupiter's  satellites,  and  those  in  which  terrestrial 
signals  are  employed. 

86.  Longitude  by  Transportation  of  Timepiece. 

In  this  method  the  error  of  the  watch  or  chronometer  with 
reference  to  the  first  meridian  is  found  by  observing  the  local 
time  at  the  first  station.  The  rate  of  the  timepiece  should  be 
determined  by  making  another  observation  at  the  same  place 
at  a  later  date.  The  timepiece  is  then  carried  to  the  second 
station  and  its  error  determined  with  reference  to  this  meridian. 
If  the  watch  runs  perfectly  the  two  watch  corrections  will 
differ  by  just  the  difference  in  longitude.  Assume  that  the  first 
observation  is  made  at  the  easterly  station  and  the  second  at 
the  westerly  station.  To  correct  for  rate,  let  r  be  the  daily 
rate  in  seconds,  +  when  losing  —  when  gaining,  c  the  watch 
correction  at  the  east  station,  cf  the  watch  correction  at  the 
west  station,  d  the  number  of  days  between  the  observations, 

139 


140  PRACTICAL  ASTRONOMY 

and  T  the  watch  reading  at  the  second  observation.  Then  the 
difference  in  the  longitude  is  found  as  follows : 

Local  time  at  W.  station  =  T  +  cf 

Local  time  at  E.  station  =  T  +  c  +  dr 

Diff.  in  time  =  Diff.  in  Long.  =  c+  dr  —  cr .  [100] 

The  same  result  will  be  obtained  if  the  stations  are  occupied 
in  the  reverse  order. 

If  the  error  of  a  mean-time  chronometer  or  watch  is  found 
by  star  observations,  it  is  necessary  t9  know  the  longitudes 
accurately  enough  to  correct  the  sun's  right  ascension.  If  a 
sidereal  chronometer  is  used  and  its  error  found  on  L.  S.  T.  this 
correction  is  rendered  unnecessary. 

In  order  to  obtain  a  check  on  the  rate  of  the  timepiece  the 
observer  should,  if  possible,  return  to  the  first  station  and  again 
determine  the  local  time.  If  the  rate  is  uniform  the  error  in 
its  determination  will  be  eliminated  by  taking  the  mean  of  the 
results.  This  method  is  not  as  accurate  as  the  telegraphic 
method,  but  if  several  chronometers  are  used  and  several  round 
trips  between  stations  are  made  it  will  give  good  results.  It  is 
useful  at  sea  and  in  exploration  surveys. 

Example. 

Observations  for  local  mean  time  at  meridian  A  indicate 
that  the  watch  is  15™  40*  slow.  At  a  point  B,  west  of  A,  the 
watch  is  found  to  be  14™  ios  slow  on  local  mean  time.  The 
watch  is  known  to  be  gaining  8s  per  day.  The  second  obser- 
vation is  made  48  hours  after  the  first.  The  difference  in  longi- 
tude is  therefore 

+  I5m4os  -  2  X  8*  -  i4m  ios  =  im  14s. 
The  meridian  B  is  therefore  im  14*  or  18'  30"  west  of  meridian  A. 

87.     Longitude  by  the  Electric   Telegraph. 

In  the  telegraphic  method  the  local  sidereal  time  is  accurately  determined  by 
star  transits  observed  at  each  of  the  stations.  The  observations  are  made 
with  large  portable  transits  and  are  recorded  on  chronographs  which  are  connected 


OBSERVATIONS  FOR  LONGITUDE  141 

with  break-circuit  chronometers.  The  stars  observed  are  chosen  in  such  a  manner 
as  to  determine  the  errors  of  the  instruments  so  that  these  may  be  eliminated 
from  the  results  as  completely  as  possible.  Some  of  the  stars  are  slowly  moving 
(circumpolar)  stars  and  others  are  more  rapidly  moving,  stars  near  the  zenith; 
a  comparison  of  these  two  makes  it  possible  to  compute  the  azimuth  of  the  line 
of  collimation.  Half  of  the  stars  are  observed  with  the  instrument  in  one  position, 
half  in  the  reversed  position;  this  determines  the  error  in  the  sight  line.  The 
inclination  error  is  measured  with  the  striding  level. 

After  the  corrections  to  the  two  chronometers  have  been  accurately  determined 
the  two  chronographs  are  switched  into  the  main-line  circuit  and  signals  are  sent 
by  breaking  the  circuit  a  number  of  times  by  pressing  a  telegraph  key.  These 
signals  are  recorded  on  both  chronographs.  In  order  to  eliminate  the  error  due 
to  the  time  required  in  transmitting  a  signal,*  these  signals  are  sent  first  in  one 
direction  (E-W)  and  afterward  in  the  opposite  direction  (W-E).  In  this 
manner  the  transmission  time  is  eliminated,  provided  it  is  constant.  The  personal 
errors  of  the  observers  are  eliminated  by  the  observers  exchanging  places  in  the 
middle  of  the  series;  i.e.,  the  above  operation  would  be  repeated  for  about  five 
nights  with  the  observers  in  one  position  and  then  for  five  nights  after  the  observers 
have  exchanged  positions.  After  all  of  the  observations  have  been  corrected  for 
instrumental  errors,  and  the  error  of  the  chronometer  on  local  sidereal  time  is 
known,  each  signal  sent  over  the  main  line  will  be  found  to  correspond  to  a  certain 
instant  of  sidereal  time  at  the  east  station  and  a  different  instant  of  sidereal  time 
at  the  west  station.  This  difference  is  the  difference  in  longitude.  The  mean  of 
all  these  values  is  the  final  difference  free  from  errors  in  transmission  time  and  per- 
sonal errors.  By  this  method  the  difference  in  longitude  may  be  determined  with 
an  error  of  perhaps  10  to  20  feet  on  the  earth's  surface. 

88.    Longitude  by  Transit  of  the  Moon. 

A  method  which  is  easily  used  with  the  surveyor's  transit  and  which,  although 
not  precise,  may  be  of  use  in  exploration  surveys,  is  that  of  determining  the  moon's 
right  ascension  by  observing  its  transit  over  the  meridian.  The  right  ascension 
of  the  moon's  centre  is  tabulated  in  the  Nautical  Almanac  for  every  hour  of 
Greenwich  Mean  Time;  hence,  if  the  right  ascension  can  be  determined,  the 
Greenwich  time  can  be  computed.  A  comparison  of  this  with  local  time  gives 
the  longitude. 

The  observation  consists  in  placing  the  instrument  in  the  plane  of  the  meridian 
and  noting  the  time  of  transit  of  the  bright  limb  f  of  the  moon  and  also  of  several 
stars  whose  declinations  are  nearly  the  same  as  that  of  the  moon.  The  observed 
time  interval  between  the  moon's  transit  and  that  of  a  star  (reduced  to  sidereal 
time  if  necessary),  added  to  or  subtracted  from  the  star's  right  ascension,  gives 
the  right  ascension  of  the  moon's  limb.  A  value  of  the  right  ascension  is  obtained 

*  In  a  test  made  in  1905  it  was  found  that  the  time  signal  sent  from  Washington 
reached  Lick  Observatory,  Mt.  Hamilton,  Cal.,  in  os.o5. 

f  The  table  of  moon  culmination  in  the  Ephemeris  shows  which  limb  (I  or  II) 
may  be  observed.  See  also  note,  p.  143. 


142  PRACTICAL  ASTRONOMY 

from  each  star  and  the  mean  value  used.  To  obtain  the  right  ascension  of  the 
centre  of  the  moon  it  is  necessary  to  apply  to  the  right  ascension  of  the  limb  a 
correction  taken  from  the  Ephemeris  called '"  sidereal  time  of  semidiameter  passing 
meridian."  In  computing  this  correction  the  increase  in  the  right  ascension 
during  this  short  interval  has  been  allowed  for;  so  the  result  is  not  the  right  ascen- 
sion of  the  centre  at  the  instant  of  the  observation,  but  its  right  ascension  at  the 
instant  of  the  transit  of  the  centre  over  the  meridian.  If  the  west  limb  was 
observed  this  correction  must  be  added;  if  the  east  limb  was  observed  it  must  be 
subtracted.  The  result  is  the  right  ascension  of  the  centre  at  the  instant  of 
transit,  which  is  also  the  local  sidereal  time  at  that  instant.  Then  the  Greenwich 
Mean  Time  corresponding  to  this  instant  is  found  by  interpolating  in  the  table 
giving  the  moon's  right  ascension  for  every  hour.  To  obtain  the  G.  M.  T.  by 
simple  interpolation  find  the  next  less  right  ascension  in  the  table  and  the  "diff. 
for  im  "  on  the  same  line;  subtract  the  tabular  right  ascension  from  the  given 
right  ascension  (found  from  the  observation)  and  divide  this  difference  by  the 
"  diff.  for  iw."  The  result  is  the  number  of  minutes  and  decimals  of  minutes 
to  be  added  to  the  hour  of  G.  M.  T.  opposite  the  tabular  right  ascension  used. 
If  the  "  diff.  for,  im  "  is  varying  rapidly  it  will  be  more  accurate  to  interpolate  as 
follows.  Interpolate  between  the  two  values  of  the  "  diff.  for  im  "  and  obtain  a 
"  diff.  for  im  "  which  corresponds  to  the  middle  of  the  interval  over  which  the  inter- 
polation is  carried.  In  observations  made  with  the  surveyor's  transit  this  more 
accurate  interpolation  is  seldom  necessary. 

In  order  to  compare  the  Greenwich  time  with  the  local  time  it  is  necessary  to 
convert  the  G.  M.  T.  just  obtained  into  the  corresponding  instant  of  Greenwich 
Sidereal  Time.  The  difference  between  this  and  the  local  sidereal  time  is  the  longi- 
tude from  Greenwich. 

In  preparing  for  observations  of  the  moon's  transit  the  Nautical  Almanac 
should  be  consulted  (Table  of  Moon  Culminations)  to  see  whether  an  observation 
can  be  made  and  to  find  the  approximate  time  of  transit.  The  civil  date  should 
be  converted  into  astronomical  before  entering  the  Almanac.  The  time  of  the 
moon's  transit  may  be  taken  from  the  column  headed  "  Mean  time  of  transit  " 
and  corrected  for  longitude,  or  it  may  be  computed  from  the  approximate  right 
ascension.  The  altitude  of  the  moon  should  be  computed  as  for  a  star,  and 
in  addition  the  parallax  correction  should  be  applied.  The  moon's  parallax 
is  so  large  that  the  moon  probably  would  not  be  in  the  field  of  the  telescope 
at  all  if  this  correction  were  neglected.  The  horizontal  parallax  multiplied 
by  the  cosine  of  the  altitude  is  the  correction  to  be  applied;  the  moon  will 
appear  lower  than  it  would  if  seen  at  the  centre  of  the  earth,  so  the  correction  is 
negative. 

Since  the  moon  increases  its  right  ascension  about  2s  in  every  ITO  of  time  it  is 
evident  that  any  error  in  determining  the  right  ascension  will  produce  an  error 
about  thirty  times  as  great  in  the  longitude,  so  that  this  method  cannot  be  made  to 
give  very  precise  results. 

Following  is  an  example  of  an  observation  for  longitude  by  the  method  of 
moon  culminations  made  with  an  engineer's  transit. 


OBSERVATIONS  FOR  LONGITUDE 


143 


Example. 

Observed  transit  of  Moon  on  Jan.  9,  1900,  for  longitude.  Moon's  west  limb 
passed  cross  hair  at  6h  $gm  37s.?;  5  Ceti  passed  at  jh  02™  57s.o;  and  7  Ceti  passed 
at  7^  o6TO  42s.o. 


5  Ceti 
Moon's  Limb 


Sid.  int. 
R.  A.  5  Ceti 

R.  A.  M.'sLimb 


jh    O2m     57*.o    7  Ceti 

6     59        37  .  7     Moon's  Limb 

03m     i9s  .  3 
•55 


°3 
34 


19  . 85  Sid.  int. 
23  .  02  R.  A.  7  Ceti 


7h   06* 
6      59 


42-* .  o 
37  -7 


07™     04s.  3 
i  .16 

O7TO     O5S.  46 
38       08  .77 


=  2h    31™   03*.  17  R.A.M.'sLimb  =  2h    31™     03*.  31 

2        31          03    .31 


Mean  = 

Time  of  s.  d.  passing  merid.  = 
R.  A.  M.'s  Centre  =  2 


O3S.  24 
08  .  86 


32™    12".  10  =  L.  S.  T. 


From  the  Nautical  Almanac. 


G.  M.  T. 


12 


R.  A.  Moon 

2h  29™    55s 

2       32          12 


77 

32 


Diff.  for 

2.  2748 

2.  2767 


32™    12s.  10 

29     55  -77 

2m  l6s.  33  =  i36s.33      log  2.  13459    G.  M.  T.      =  u*5Q»»  54s- 42 
Interpolated  Diff.  im          =2.2757       logo.  35711    R.  A.  M.S.  =  19    14      15.92 

i      58  .  26 
M.  T.  Interval  =  59™.  907    log  i.  77748 

G.  S.  T.       =    7h  i6m  o8s.  60 
G.  M.  T.  =  nh  59m  54s.  42  L.  S.  T.       =2    32      12  .  10 

Long.  W.     =    4h  43m  56s.  50 

NOTE.  It  has  already  been  stated  that  the  moon  moves  eastward  on  the  celes- 
tial* sphere  at  the  rate  of  about  13°  per  day;  as  a  result  of  this  motion  the  time  of 
meridian  passage  occurs  about  51"*  later  (on  the  average)  each  day.  On  account 
of  the  eccentricity  of  its  orbit,  however,  the  actual  retardation  may  vary  consid- 
erably from  the  mean.  The  moon's  orbit  is  inclined  at  an  angle  of  about  5°  08' 
to  the  plane  of  the  earth's  orbit.  The  line  of  intersection  of  these  two  planes  ro- 
tates in  a  similar  manner  to  that  described  under  the  precession  of  the  equinoxes, 
except  that  its  period  is  only  19  years.  The  moon's  maximum  declination,  there- 
fore, varies  from  23°  27'+  5°  08'  to  23°  27'  —  5°  08',  that  is,  from  28°  35'  to  18°  19', 


144 


PRACTICAL  ASTRONOMY 


according  to  the  relative  position  of  the  plane  of  the  moon's  orbit  and  the  plane  of 
the  equator.  The  rapid  changes  in  the  relative  position  of  the  sun,  moon,  and  earth, 
and  the  consequent  changes  in  the  amount  of  the  moon's  surface  that  is  visible 
from  the  earth,  cause  the  moon  to  present  the  different  aspects  known  as  the 
moon's  phases.  Fig.  58  shows  the  relative  positions  of  the  three  bodies  at  several 


€  ~ 

First  Quarter 


o 


FIG.  58.    THE  MOON'S  PHASES 


different  times  in  the  month.    The  appearance  of  the  moon  as  seen  from  the  earth 
is  shown  by  the  figures  around  the  outside  of  the  diagram. 

It  may  easily  be  seen  from  the  diagram  that  at  the  time  of  first  quarter  the 
moon  will  cross  the  meridian  at  about  6  P.M.;  at  full  moon  it  will  transit  at  mid- 
night; and  at  last  quarter  it  will  transit  at  about  6  A.M.  Although  the  part  of  the 
illuminated  hemisphere  which  can  be  seen  from  the  earth  is  continually  changing, 
the  part  of  the  moon's  surface  that  is  turned  toward  the  earth  is  always  the  same, 
because  the  moon  makes  but  one  rotation  on  its  axis  in  one  lunar  month.  Nearly 
half  of  the  moon's  surface  is  never  seen  from  the  earth. 


OBSERVATIONS   FOR  LONGITUDE  145 

Longitude  by  Time  Signals. 

If  it  is  desired  to  obtain  an  approximate  longitude  for  any  pur- 
pose this  may  be  done  in  a  very  simple  manner  provided  the 
observer  is  able  to  obtain  the  standard  time  at  some  telegraph 
station  as  given  by  the  noon  signals.  He  may  determine  his 
local  mean  time  by  any  of  the  preceding  methods  (Chapter  XI) . 
The  difference  between  the  local  time  and  the  standard  time  by 
telegraph  is  the  correction  to  be  applied  to  the  longitude  of  the 
standard  meridian  to  obtain  the  longitude  of  the  observer. 

Example. 

Altitude  of  sun,  27° 44' 35";  latitude,  42°22'N.;  declination,  iQ0oo'oo"N.; 
equation  of  time,  +3m488.8;  watch  reading,  4h  i8m  i3*.8.  From  these  data  the 
local  mean  time  is  found  to  be  4A33m43s.Q,  making  the  watch  15™  30/1  slow. 
By  comparison  with  the  telegraph  signal  at  noon  the  watch  is  found  to  be  6s  fast 
of  Eastern  Standard  Time.  The  longitude  is  then  computed  as  follows:  — 

Correction  to  L.  M.  T.  =   +15™  30". i 

"     E.  S.  T.  =    -oo    06  .o 

Difference  in  Longitude  =        I5m  ^5*  r 

=  3°  54'  oi".s 

Longitude  =  75°  -  3°  s^'.o  =  7i°o6'.o  West 
Questions  and  Problems 

1.  Compute  the  longitude  from   the  following  observed  transits:  8  Aquarii, 
$h  i6m  04*;  TT  Aquarii,  $h  24™  40*;  moon's  W.  limb,  5*  32"*  27*;  X  Aquarii,  5*  $im  47*. 
R.  A.  6  Aquarii  =  22h  nm  27*.6;  R.  A.  TT  Aquarii,  =  22h  20™  O48.6;  R.  A.  X  Aquarii 
=  22h  47"*  i8*.3;  sidereal  time  of  semidiameter  passing  meridian  =  6o*.3;  at 
G.  M.  T.  10*,  moon's  R.  A.  =  22*  27™  53*.3;  diff.  for  im  =  i'.g8oo;  R.  A.  mean 
sun  at  G.  M.  N.  =  i6h  38™  28s .o. 

2.  When  employing  the  lunar  method  can  the  longitude  be  computed  by  com- 
paring G.  M.  T.  and  L.  M.  T.? 

3.  Which  limb  can  be  observed  in  a  P.  M.  observation  of  a  moon  culmination? 

4.  At  about  what  time  (mean  local)  will  the  moon  transit  when  it  is  at  first 
quarter? 

5.  Altitude  of  sun,  57°  15'  36";  latitude,  42°  22'  N.;  declination,  18°  s8'.6  N.; 
equation  of  time,  +3™  49";  watch,  ih  29™  o88,  P.M.     Error  of  watch  on  E.  S.  T. 
by  time  signal  at  noon,  ios  fast.     Compute  the  longitude. 


CHAPTER  XIII 
OBSERVATIONS  FOR  AZIMUTH 

89.  Determination  of  Azimuth. 

The  determination  of  the  azimuth  of  a  line  is  of  frequent 
occurrence  in  the  practice  of  the  surveyor,  and  is  the  most 
important  to  him  of  all  the  astronomical  problems.  On  account 
of  the  high  altitudes  of  the  objects  observed,  as  compared  with 
those  observed  in  surveying,  the  adjustments  of  the  instru- 
ment and  the  elimination  of  errors  are  of  unusual  importance 
in  these  observations.  All  of  the  precautions  mentioned  in 
Chapter  VIII  in  regard  to  stability  of  the  instrument,  etc., 
should  be  carefully  observed:  the  instrument  should  be  allowed 
to  stand  for  some  time  before  observations  are  begun;  temper- 
ature changes  from  any  source,  such  as  heat  from  the  lamp  or 
from  the  hand,  are  to  be  avoided;  the  clamps  and  tangent  screws 
should  be  used  with  the  same  care  as  in  triangulation  work  if 
the  greatest  accuracy  is  desired  in  the  results. 

90.  Azimuth  Mark. 

When  the  observation  is  made  at  night  it  is  frequently  incon- 
venient to  sight  directly  at  the  object  whose  azimuth  is  to  be 
determined ;  it  is  necessary  in  such  cases  to  determine  the  azimuth 
of  a  special  mark  called  the  azimuth  mark,  which  can  be  seen 
both  at  night  and  in  daylight,  and  then  to  measure  the  angle 
between  this  mark  and  the  first  object  during  the  day.  The 
azimuth  mark  usually  consists  of  a  lamp  set  inside  of  a  box 
having  a  small  hole  cut  in  the  side,  through  which  the  light 
may  shine.  The  size  of  the  opening  should  be  determined  by 
the  distance  of  the  mark;  for  accurate  work  it  should  subtend 
an  angle  not  greater  than  about  o".5  to  i".o.  If  possible, 
the  mark  should  be  a  mile  or  more  distant,  so  that  the  focus 
of  the  telescope  will  not  have  to  be  altered  when  changing  from 

146 


OBSERVATIONS   FOR  AZIMUTH  147 

the  star  to   the  mark.     It  is  frequently  necessary,   however, 
to  set  the  mark  nearer  on  account  of  the  topographic  and  other 
conditions. 
91.  Azimuth  of  Polaris  at  Elongation. 

The  simplest  method  of  determining  the  direction  of  the 
meridian  with  accuracy  is  by  means  of  an  observation  of  the 
polestar,  or  any  other  close  circumpolar,  when  it  is  at  its  greatest 
elongation.  (See  Art.  19,  p.  31.)  The  appearance  of  the 
constellations  at  the  time  of  this  observation  on  Polaris  may 
be  seen  by  referring  to  Fig.  49.  When  the  polestar  is  west  of 
the  pole  the  Great  Dipper  is  on  the  right  and  Cassiopeia  on  the 
left.  The  exact  time  of  elongation  may  be  found  by  comput- 
ing the  sidereal  time  when  the  star  is  at  elongation,  and  convert- 
ing this  into  mean  solar  time  (local  or  standard)  by  the  methods 
of  Arts.  34  and  35.  To  find  the  sidereal  time  of  elongation  first 
compute  the  hour  angle  Pe  by  Equa.  [34]  and  then  convert  it 
into  time.  If  western  elongation  is  desired,  then  Pe  is  the  hour 
angle;  if  eastern  elongation  is  desired,  then  24*  —  Pe  is  the  true 
hour  angle.  The  sidereal  time  is  then  found  by  Equa.  [37],  p.  48. 
An  average  value  of  Pe  for  Polaris  for  latitudes  between  30°  and 
50°  is  about  5A  55m;  this  is  sufficiently  accurate  for  computing 
the  time  of  elongation  for  many  purposes.  Approximate  values 
of  the  times  of  elongation  of  Polaris  may  be  taken  from  Table  V. 

Example. 

Find  the  Eastern  Standard  time  of  Western  Elongation  of 
Polaris  on  April  6,  1904,  in  lat.  42°  21';  long.  4^  44™  i8s  W. 
The  right  ascension  is  ih  25™  48s-3;  the  declination  is  +  88°  47' 
43".6;  the  sun's  right  ascension  at  G.  M.  N.  is  oh  57™  2 2 '.44. 

log  tan  L  =9-95977  ?  =  5h    55W    36s.  5  S   =  7h    i9TO    24*.  8 

log  tan  D  =  i.  67723  R  =  i      23       48  . 3  Rs  =  o     58      09  .  i 

log  cos  P  =8.28254  S  =  7A    igm    24*.  8  6h    21™    158.7 

P  =  88°  54'  07"  +  C  =      -  i       02  . 5 

=  5h  5Sm  36s.  5 

M.  L.  T.  =  6h    20™    i3s .  2 

15      42 
E.  S.T.   =  6h  04™    31*.  2 


148  PRACTICAL  ASTRONOMY 

The  transit  should  be  set  in  position  half  an  hour  or  so 
before  elongation.  The  star  is  bisected  by  the  vertical  cross 
hair,  and  as  it  moves  out  toward  its  greatest  elongation  its 
motion  is  followed  by  means  of  the  tangent  screw  of  the  upper 
or  the  lower  plate.  Near  the  time  of  elongation  the  star  will 
appear  to  move  almost  vertically,  so  that  no  motion  in  azimuth 
can  be  detected  for  five  minutes  or  so  before  or  after  elonga- 
tion. About  5m  before  elongation,  centre  the  plate  levels,  set 
the  cross  hair  carefully  on  the  star,  lower  the  telescope  with- 
out disturbing  its  azimuth,  and  set  a  stake  or  a  mark  carefully 
in  line  at  a  distance  of  several  hundred  feet  north  of  the  transit. 
Reverse  the  telescope,  recentre  the  levels  if  necessary,  bisect 
the  star  again,  and  set  another  point  beside  the  first  one.  If 
there  are  errors  of  adjustment  the  two  points  will  not  coincide; 
the  mean  of  the  two  is  the  true  point.  The  angle  between  the 
meridian  and  the  line  to  the  stake  (the  star's  azimuth)  is  found 
by  the  equation 

sin  Z  =  sin  p  sec  L  [35] 

where  Z  is  the  azimuth  from  the  north;  p,  the  polar  distance  of 
the  star;  and  L,  the  latitude  of  the  place.  L  does  not  have  to 
be  known  with  great  precision;  an  error  of  i'  in  L  produces 
only  about  i"  error  in  the  azimuth  of  Polaris  for  latitudes  within 
the  United  States.  The  above  method  may  be  applied  to  any 
close  circumpolar  star.  For  Polaris,  whose  polar  distance  is 
about  i°  10',  it  is  usually  accurate  enough  to  use  the  formula 

Z"  =  p"  sec  L,  [101] 

in  which  Z"  and  p"  are  expressed  in  seconds  of  arc.  This 
computed  angle  may  be  laid  off  in  the  proper  direction  with  a 
transit  (by  daylight),  using  the  method  of  repetitions,  or  with 
a  tape,  by  means  of  a  perpendicular  offset  calculated  from  the 
measured  distance  to  the  stake  and  the  calculated  azimuth 
angle.  (Fig.  59.)  The  result  is  the  true  north  and  south  line. 
It  is  often  desirable  to  measure  the  horizontal  angle  between 


OBSERVATIONS   FOR  AZIMUTH 


149 


c 

Star 


the  star  at  elongation  and  some  fixed  point  instead  of  marking 
the  meridian  itself.  On  account  of  the  slow  change  in  azimuth 
there  is  ample  time  to  measure  several  repetitions  before  the 
error  in  azimuth  amounts  to  more  than  i"  or 
2"  *  The  errors  of  adjustment  of  the  transit 
will  be  eliminated  if  half  of  the  angles  are 
taken  with  the  telescope  erect  and  half  in- 
verted. The  plate  levels  should  be  recentred 
for  each  position  of  the  instrument  before  the 
measurements  are  begun  and  while  the  telescope 
is  pointing  toward  the  star. 

Example. 

Compute  the  azimuth  of  Polaris  at  greatest 
elongation  on  April  6,  1904,  in  latitude  42° 
21'  N.  The  declination  of  the  star  for  the  given 
date  is  +  88°  47'  43". 6. 

log  sin  p  =  8. 32267 
log  sec  L  =  o.  13133 


log  sinZ  =  8.45400 

Z  =1°  3/47".  9 


FIG.  59 


By  using  the  angles  in  place  of  the  sines,  neglecting  fractions 
of  a  second,  the  following  result  is  obtained : 

P  =  4336" 

log  p  =  3.  6371 
log  sec  L  =  o.  1313 


logZ"  =  3-7684 
Z"  =  5867 


92.   Observations  Near  Elongation. 

If  the  observation  is  made  on  Polaris  at  any  time  within  half 
an  hour  of  elongation,  the  azimuth  of  the  star  at  each  pointing 


*  In  latitude  40°  the  azimuth  changes  about  i'  in  half  an  hour  before  or  after 
elongation;  the  change  in  azimuth  varies  approximately  as  the  square  of  the  time 
from  elongation. 


PRACTICAL  ASTRONOMY 


of  the  telescope  may  be  reduced  to  its  value  at  elongation, 
provided  the  time  is  known.     The  formula  for  this  reduction  is 

C  =  112.5  X  3600  X  sin  i"  X  tanZe  X  (T  -  Te)2*    [102] 

in  which  Ze  is  the  azimuth  at  elongation;  T,  the  observed  time; 
Te,  the  time  of  elongation;  and  C,  the  correction  in  seconds  of 


*  For  the  rigorous  demonstration  of  this  for- 
mula, which  is  rather  complex,  see  Doolittle's 
Practical  Astronomy.  The  following  proof, 
although  inexact,  gives  substantially  the  same 
result.  In  Fig.  60,  5  is  the  position  of  Polaris 
and  E  its  position  when  at  greatest  elongation, 
the  angle  SPE,  or  i,  being  not  greater  than, 
about  8°.  In  the  triangle  SPM , 

tan  M P  =  tan  PS  cos  SPM 
Since  the  arcs  are  small,  we  may  put 


FIG.  60 


'    or 


MP  =  SP  cos  5PM, 
MP  =  p  cos  i. 
EM  =  EP  -  PM 

=  p  —  p  cos  i. 


Replacing  cos  i  by  the  series  i \-  • 


In  the  triangle  ZME,  £E  =  90°,  and  ZM  =  ZP  (nearly),  whence 


sin  M ZE  = 


cos  L 


(nearly), 

2 


=  sin  Ze  X  -i 

in  which  Ze  is  the  azimuth  at  elongation.     Replacing  sin  MZE  by  its  arc  in  seconds 
(C")  and  reducing  i  to  minutes  of  time, 


C"  =  ^  X  sin  i"  X  (6o)2  X  (i5)2  X  sinZe. 


[103) 


Replacing  sin  Ze  by  tan  Ze  produces  an  error  of  only  about  o" '.  02  for  Polaris  in 
latitude  40°  and  reduces  [103]  to  [102]. 


OBSERVATIONS  FOR  AZIMUTH  151 

arc.  T  —  Te  must  be  in  minutes  of  (sidereal)  time.  The 
factor  112.5  X  3600  X  sin  i"  X  tanZe  may  be  computed,  and 
then  all  observations  made  at  the  same  place  at  about  the  same 
date  may  be  reduced  by  multiplying  the  square  of  the  time  in- 
tervals in  minutes  by  the  factor  computed.  Table  VII  gives 
values  of  the  factor  for  values  of  Ze  ranging  from  i°  to  2°. 
These  corrections  will  also  be  found  in  Table  Via  at  the  end 
of  the  Nautical  Almanac. 

Example. 

Three  repetitions  of  the  angle  between  Polaris  at  western  elongation  and  a 
mark  supposed  to  be  on  the  meridian,  April  6,  1904.  Lat.  42°  21';  long.  71°  04'. 5  W. 
The  observed  times  are  6h  28™  30*,  6h  31™  20s  and  6h  34"*  20s.  First  reading  of 
vernier  =  o°  oo';  last  reading  of  vernier  =  4°  51'  oo".  The  R.  A.  of  Polaris  — 
XA  23m  48s.3;  its  declination  =  +  88°  47'  43". 6.  R.  A.  Mean  Sun  at  G.  M.  N.  = 
cf1  57™  22s. 44. 

From  this  data  the  Eastern  time  of  elongation  is  found  to  be  6^  O4m  3is.2. 
The  intervals  (T  -  Te}  are  23™  58S.8,  26™  48*.8  and  29™  48S.8.  The  azimuth 
of  the  star  at  elongation  is  i°  37'  48".  From  Table  VII  the  factor  is  found  to 
be  .0559.  The  resulting  corrections  are  32",  40"  and  50".  Adding  these  to  the 
third  reading,  the  sum  is  4°  53'  02".  One  third  of  this  is  i°  37'  41",  the  measured 
angle  between  the  mark  and  the  star  at  elongation.  The  meridian  mark  is  there- 
fore 7"  west  of  north,  according  to  this  observation. 

93.   Azimuth  by  an  Altitude  of  the  Sun. 

In  order  to  determine  the  azimuth  of  a  line  by  means  of  an 
observation  on  the  sun  the  instrument  should  be  set  up  over 
one  of  the  points  marking  the  line  and  carefully  levelled.  The 
plate  vernier  is  first  set  at  o°  and  the  vertical  cross  hair  sighted 
on  the  other  point  marking  the  line.  The  colored  shade  glass 
is  then  screwed  on  to  the  eyepiece,  the  upper  clamp  loosened, 
and  the  telescope  turned  toward  the  sun.  The  sun's  disc  should 
be  sharply  focussed  before  beginning  the  observations.  In 
making  the  pointings  on  the  sun  great  care  should  be  taken 
not  to  mistake  one  of  the  stadia  hairs  for  the  middle  hair.  If 
the  observation  is  to  be  made,  say,  in  the  forenoon  (in  the 
northern  hemisphere),  first  set  the  cross  hairs  so  that  the  ver- 
tical hair  is  tangent  to  the  right  edge  of  the  sun  and  the  hori- 
zontal hair  cuts  off  a  small  segment  at  the  lower  edge  of  the 


152 


PRACTICAL  ASTRONOMY 


disc.     (Fig.  6 1.)*    The  arrow  in  the  figure  shows  the  direction  of 
the  sun's  apparent  motion.     Since  the  sun  is  now  rising  it  will 
in  a  few  seconds  be  tangent  to  the  horizontal  hair.     It  is  only 
necessary  to  follow  the  right  edge  by  means  of  the  upper  plate 
tangent  screw  until  both  cross  hairs  are 
tangent.     At  this  instant,  stop  following 
the  sun's  motion  and  note  the  time.     If 
it  is  desired  to  determine  the  time  accu- 
rately, so  that  the  watch  correction  may  be 
found  from  this  same  observation,  it  can 
be  read  more  closely  by  a  second  observer. 
FIG.  61.    POSITION  or       Both    the   horizontal    and   the    vertical 
SUN'S  Disc  A  FEW  SECONDS  circles  are  read,  and  both  angles  and  the 
BEFORE  OBSERVATION       time  are  recorded.     The  same  observa- 

(A.  M.  Observation  in  Northern        .  - 

Hemisphere.)  tion    may    be    repeated    three    or   four 

times  to  increase  the  accuracy.      The  instrument  should  then 
be  reversed  and  the  set  of  observations  repeated,  except  that 
the  horizontal  cross  hair  is  set  tangent  to 
the  upper  edge  of  the  sun  and  the  ver- 
tical cross  hair  cuts  a  segment  from  the 
left  edge   (Fig.  62).      The  same  number 
of   pointings    should   be    taken   in    each 
position  of  the   instrument.      After   the 
pointings  on  the  sun  are  completed  the 
telescope  should  be  turned  to  the  mark      FlG  62     pOSITION  OF 
again  and  the  vernier    reading    checked.   SUN'S  Disc  A  FEW  SECONDS 
If  the  transit  has  a  vertical  arc  only,  the       BEFORE  OBSERVATION 
telescope  cannot  be  used  in  the  reversed    (A' M' o^g^)Northeri1 
position  and  the  index  correction  must  therefore  be  determined. 
If  the  observation  is  to  be  made  in  the  afternoon  the  positions 
will  be  those  indicated  in  Fig.  63.! 

*  In  the  diagram  only  a  portion  of  the  sun's  disc  is  visible;  in  a  telescope  of  low 
power  the  entire  disc  can  be  seen. 

t  It  should  be  kept  in  mind  that  if  the  instrument  has  an  inverting  eyepiece 
the  direction  of  the  sun's  apparent  motion  is  reversed.  If  a  prism  is  attached  to 
the  eyepiece,  the  upper  and  lower  limbs  of  the  sun  are  apparently  interchanged, 
but  the  right  and  left  limbs  are  not. 


OBSERVATIONS  FOR  AZIMUTH 


153 


In  computing  the  azimuth  it  is  customary  to  neglect  the  cur- 
vature of  the  sun's  path  during  the  short  interval  between  the 
first  and  last  pointings,  unless  the  series  extends  over  a  longer 
period  than  is  usually  required  to  make  such  observations. 
If  the  observation  is  taken  near  noon  the  curvature  is  greater 
than  when  it  is  taken  near  the  prime  vertical.  The  mean  of 
the  altitudes  and  the  mean  of  the  horizontal  angles  are  assumed 
to  correspond  to  the  position  of  the  sun's  centre  at  the  instant 
shown  by  the  mean  watch  reading.  The  mean  altitude  read- 
ing corrected  for  refraction  and  parallax  is  the  true  altitude  of 


FIG.  63.    POSITIONS  or  SUN'S  Disc  A  FEW  SECONDS  BEFORE  OBSERVATION 

(P.  M.  Observation  in  Northern  Hemisphere.) 

the  sun's  centre.  The  azimuth  is  then  computed  by  any  one 
of  the  formulae  on  page  34.  The  resulting  azimuth  combined 
with  the  mean  horizontal  circle  reading  gives  the  azimuth  of 
the  mark.  Five-place  logarithmic  tables  will  give  the  azimuth 
within  5"  to  10",  which  is  as  great  a  degree  of  precision  as  can 
be  expected  in  this  method. 

If  for  any  reason  only  one  limb  of  the  sun  has  been  observed, 
the  azimuth  observed  may  be  reduced  to  the  centre  of  the  sun 
by  applying  the  correction  6*  sec  h,  where  S  is  the  semidiameter 
and  h  is  the  altitude  of  the  centre. 

If  one  has  at  hand  a  set  of  tables  containing  log  versed  sines  (such  as  are  in- 
cluded in  railroad  engineering  tables)  the  following  formulae  will  sometimes  be 
found  useful. 

cos  (L  +  h)  +  sin  D  [104] 


and 


vers  Zs 


versZn  = 


cos  L  cos  h 

cos  (L  —  h}  —  sin  D 
cos  L  cos  h 


[105] 


154 


PRACTICAL  ASTRONOMY 


The  sum  or  difference  in  the  numerator  must  be  computed  by  natural  functions 
and  the  remainder  of  the  work  performed  by  means  of  logarithms. 
Example. 

Observation  on  Sun  for  Azimuth. 
Lat.  42°  21'  N.  Long.  4^  44™  i8sW.  Date,  Nov.  28,  1905. 


Hor.  Circle. 
Ver.  A.      B. 

238°  14'    14' 

311  48 

312  20 


Mark 

R  &  L  limbs 

R  &  L  limbs 

(instrument  reversed) 
L  &  U  limbs  312  27 
L  &  U  limbs  312  52 
Mark  238  14 


48.5 
20 


Vert.  Circle. 


Watch. 

A.  M. 


8    42     19 


26.5 
51-5 

14 


55 

68 


45 

47 


34 
34 


Mean  reading  on 

mark  238°   14'.  o 

Mean  reading  on 

sun  312     21  .  7 


Mean  =  15°  26' 
R&P=  3-3 


Mean  =  Sh  43TO  47* 


Mark  N.  of  sun 

L  =  42°  21'.  o 
h  =  15  22  .  7 
p  =  in  15  .  7 


74    07'.  7 


h  =  15°  22'.  7 


G.  M.  T  =   ih  43™  47s 
Sun's  Decl.  at  G.  M.  N.   =  -    21°   14'    54".  4 


-  26".8i  X 


.73 


-     46  .4 


2s   =  168°  59'.  4 
s  =     84°  29'.  7 

s  -  L  =       42°  08'.  7 

s  —  h    =       69    07  .  o 

s  —  p    =  —  26     46  .  o 

s   =       84     29.7 


Declination 
N.  Polar  distance 

log  sin  9.82673 
log  sin  9.97049 
log  sec  o.  04922 
log  sec  1.01804 

2)0.  86448 


=    —      21       15       40 
=  111°    I5'     40' 


log  tan  £  Zn  =  o.  43224 

\Zn=     69°    42'.  9 
Zn  =  139°    25'-  8 

Mark  N.  of  sun       =    74    07  .  7 


Bearing  of  Mark=  N  65°  18'.  i  E 

If  formula  [27]  is  employed  the  computation  would  be  arranged  as  follows: 
nat.  sin  D   =  —  .36262 
log  sin  L  = 
log  sin  h   = 

nat.  sin  L  sin  h  =       .17865 
Numerator  =  —  .54127 
log  numerator  =     9.  73342  (n) 
log  sec  L  =    o.  13133 
log  sec  h  =    0.01584 
log  cos  Zn  =    9.88059  (n) 

Zn  =  139°  2S'.8  as  before. 


9.82844 
9-42356 
9.25200 


OBSERVATIONS  FOR  AZIMUTH  155 

By  differentiating  Equa.  [13]  it  may  be  shown  that  when  the  latitude  is  greater 
than  the  sun's  declination  the  greatest  accuracy  in  the  azimuth,  so  far  as  errors 
in  altitude  are  concerned,  is  secured  when  the  sun  is  somewhere  between  the  prime 
vertical  and  the  six-hour  circle;  the  exact  position  for  maximum  accuracy  depends 
upon  the  latitude  and  upon  the  parallactic  angle.  If  an  observer  were  on  the 
equator  and  the  sun's  declination  zero,  the  motion  would  be  vertical  and  the 
change  in  azimuth  would  be  zero.  In  the  preceding  example  the  azimuth  increases 
about  i'  50"  for  an  increase  of  i'  in  the  altitude.  Errors  in  the  azimuth  due  to 
errors  in  the  assumed  value  of  the  latitude  are  a  minimum  when  the  sun  is  on  the 
six-hour  circle.  Observations  very  near  the  horizon,  however,  are  subject  to  errors 
in  the  refraction,  since  the  tabular  values  of  the  mean  refraction  may  be  largely 
in  error  for  very  low  altitudes  under  the  temperature  and  pressure  conditions 
existing  at  the  time  of  the  observation.  The  general  rule  therefore  is  to  avoid 
observations  near  the  meridian  and  also  those  that  are  within  10°  of  the  horizon. 

If  it  is  desired  to  compute  the  hour  angle  of  the  sun  from  the  same  observations 
used  in  determining  the  azimuth,  it  may  be  found  by  formula  [19],  in  which  case 
no  new  logarithms  have  to  be  taken  from  the  tables;  or  it  may  be  found  by  equation 
[12]  as  follows: 

log  sin  Z  =9.81317  L.  A.  T.  =  9^     ioTO5o8.3 

log  cos  h  =9.98416  Eq.  T.      =          12     02  .8 

log  sec  0  =  0.03061  L.M.T.=  8*    58-  47'  -S 
log  sin  P  =  9.82794  I5    42 


Watch     =8      43    47  •  2 
Watch  fast  =  4i8.  7 

94.  Azimuth   by   an   Altitude   of    a   Star   near    the    Prime 
Vertical. 

The  method  described  in  the  preceding  article  applies  equally 
well  to  an  observation  on  a  star,  except  that  the  star's  image 
is  bisected  with  both  cross  hairs  and  the  parallax  and  semi- 
diameter  corrections  become  zero.  The  declination  of  the  star 
changes  so  little  during  one  day  that  it  may  be  regarded  as 
constant,  and  consequently  the  time  of  the  observation  is  not 
required.  Errors  in  the  altitude  and  the  latitude  may  be  par- 
tially eliminated  by  combining  two  observations,  one  on  a  star 
about  due  east  and  the  other  on  one  about  due  west. 

95.  Azimuth  Observation  on  a  Circumpolar  Star  at  any  Hour  Angle. 

The  most  precise  determination  of  azimuth  may  be  made  by  measuring  the 
horizontal  angle  between  a  circumpolar  star  and  an  azimuth  mark,  the  hour  angle 


156 


PRACTICAL  ASTRONOMY 


of  the  star  at  each  pointing  being  known.  If  the  sidereal  time  is  determined 
accurately,  by  any  of  the  methods  given  in  Chapter  XI,  the  hour  angle  of  the  star 
may  be  found  at  once  by  Equa.  [37]  and  the  azimuth  of  the  star  at  the  instant 
may  be  computed.  Since  the  close  circumpolar  stars  move  very  slowly  and 
errors  in  the  observed  times  will  have  a  small  effect  upon  the  computed  azimuth, 
it  is  evident  that  only  such  stars  should  be  used  if  precise  results  are  sought.  The 
advantage  of  observing  the  star  at  any  hour  angle,  rather  than  at  elongation,  is 
that  the  number  of  observations  may  be  increased  indefinitely  and  greater  accuracy 
thereby  secured. 

The  angles  may  be  measured  either  with  a  repeating  instrument   (like  the 
engineer's  transit)  or  with  a  direction  instrument  in  which  the  circles  are  read  with 


51  Cephei 


XIIA 


FIG.  64 

great  precision  by  means  of  micrometer  microscopes.  For  refined  work  the  instru- 
ment should  be  provided  with  a  sensitive  striding  level.  If  there  is  no  striding 
level  provided  with  the  instrument  *  the  plate  level  which  is  parallel  to  the  hori- 
zontal axis  should  be  a  sensitive  one  and  should  be  kept  well  adjusted.  At  all 
places  in  the  United  States  the  celestial  pole  is  at  such  high  altitudes  that  errors 
in  the  adjustment  of  the  horizontal  axis  and  of  the  sight  line  have  a  compara- 
tively large  effect  upon  the  results. 

The  star  chosen  for  this  observation  should  be  one  of  the  close  circumpolar  stars 
given  in  the  special  list  in  the  Nautical  Almanac.  (See  Fig.  64.)  Polaris  is  the  only 
bright  star  in  this  group  and  should  be  used  in  preference  to  the  others  when  it  is 

*  The  error  due  to  inclination  of  the  axis  may  be  eliminated  by  taking  half 
of  the  observations  direct  and  half  on  the  image  of  the  star  reflected  in  a  basin 
of  mercury. 


OBSERVATIONS   FOR  AZIMUTH  1 57 

practicable  to  do  so.  If  the  time  is  uncertain  and  Polaris  is  near  the  meridian, 
in  which  case  the  computed  azimuth  would  be  uncertain,  it  is  better  to  use  51 
Cephei*  because  this  star  would  then  be  near  its  elongation  and  comparatively 
large  errors  in  the  time  would  have  but  little  effect  upon  the  computed  azimuth. 
If  a  repeating  theodolite  or  an  ordinary  transit  is  used  the  observations  consist 
in  repeating  the  angle  between  the  star  and  the  mark  a  certain  number  of  times 
and  then  reversing  the  instrument  and  making  another  set  containing  the  same 
number  of  repetitions.  Since  the  star  is  continually  changing  its  azimuth  it 
is  necessary  to  read  and  record  the  time  of  each  pointing  on  the  star  with  the 
vertical  cross  hair.  The  altitude  of  the  star  should  be  measured  just  before  and 
again  just  after  each  half -set  so  that  its  altitude  for  any  desired  instant  may  be 
obtained  by  simple  interpolation.  If  the  instrument  has  no  striding  level  the 
cross-level  on  the  plate  should  be  recentred  before  the  second  half-set  is  begun. 
If  a  striding  level  is  used  the  inclination  of  the  axis  may  be  measured,  while  the 
telescope  is  pointing  toward  the  star,  by  reading  both  ends  of  the  bubble,  with  the 
level  first  in  the  direct  position  and  then  in  the  reversed  position. 

In  computing  the  results  the  azimuth  of  the  star  might  be  computed  for  each 
of  the  observed  times  and  the  mean  of  these  azimuths  combined  with  the  mean 
of  the  measured  horizontal  angles.  The  labor  involved  in  this  process  is  so  great, 
however,  that  the  common  practice  is  first  to  compute  the  azimuth  corresponding 
to  the  mean  of  the  observed  times,  and  then  to  correct  this  result  for  the  effect  of 
the  curvature  of  the  star's  path,  i.e.,  by  the  difference  between  the  mean  azimuth 
and  the  azimuth  at  the  mean  of  the  times.  The  formula  for  the  azimuth  is 

_  sin  P .     , 

~cosLtanL>-sinLcosP' 

The  formula  given  below,  although  not  exact,  is  sufficiently  accurate  for  all  work 
except  refined  geodetic  observations. 

Z"  =  p"  sin  P  sec  h,  [106] 

in  which  Z"  and  p"  are  in  seconds  of  arc.  In  this  formula  the  arcs  have  been 
substituted  for  their  sines.  The  precision  of  the  computed  azimuth  depends 
chiefly  upon  the  precision  with  which  h  can  be  determined.  If  the  vertical  arc 
cannot  be  relied  upon,  and  the  latitude  is  known  accurately,  the  first  formula 
may  be  preferred.  If  desired,  the  altitude  of  Polaris  may  be  computed  by  formula 
[80],  p.  no,  and  its  value  substituted  in  [106]. 

*  51  Cephei  may  be  found  by  first  pointing  on  Polaris  and  then  changing  the 
altitude  and  the  azimuth  by  an  amount  which  will  bring  51  Cephei  into  the  field. 
The  difference  in  altitude  and  in  azimuth  may  be  obtained  with  sufficient  accuracy 
by  holding  Fig.  64  so  that  Polaris  is  in  its  true  position  with  reference  to  the 
meridian  (as  indicated  by  the  position  of  8  Cassiopeia)  and  then  estimating  the 
difference  in  altitude  and  the  difference  in  azimuth.  It  should  be  remembered 
that  the  distance  of  51  Cephei  east  or  west  of  Polaris  has  nearly  the  same  ratio 
to  the  difference  in  azimuth  that  the  polar  distance  of  Polaris  has  to  its  azimuth 
at  elongation,  i.e.,  i  to  sec  L. 


158  PRACTICAL  ASTRONOMY 

96.  The  Curvature  Correction. 

If  we  let  TI,  Tz,  T3,  etc.  =  the  observed  times,  T0  =  the  mean  of  these  times, 
Zi,  Z2,  Z3,  etc,  =  corresponding  azimuths,  and  Z0  the  azimuth  at  the  instant  T0, 
then 

Zl+Z2+    '••  Zn  ^ZQ_  ^^  [a  j  I  s  (r  _  ro)2  ,  [iQ7] 

w  w 

The  quantity  in  brackets  is  the  logarithm  of  a  constant;  S  (T  —  T0)2  is  the  sum  of 
the  squares  of  the  time-intervals  (in  minutes  and  decimals)  reduced  to  sidereal 
intervals.  The  azimuth  is  therefore  computed  by  first  finding  Z0  by  Equa.  [31] 
and  then  correcting  it  by  means  of  the  last  term  of  Equa.  [107]. 

If  it  is  desired  to  express  (T  —  T0)  in  seconds  of  time  the  constant  log  becomes 
[6.73672].  When  the  star  is  near  culmination  the  curvature  correction  is  very 
small;  near  elongation  it  is  a  maximum. 

97.  The  Level  Correction. 

The  inclination  i  of  the  axis  as  determined  by  the  striding  level  is  given  in 
seconds  of  arc  by 

i  =    [(w  +  w')  -(e  +  e')}  -  ,  [108] 

4 

where  w  and  e  are  the  readings  of  the  west  and  east  ends  of  the  bubble  for  the 
direct  position,  and  w'  and  e'  are  the  same  for  the  reversed  positions,  and  d  is  the 
angular  value  of  one  division  of  the  level  scale.  The  correction  to  the  measured 
horizontal  angle  is 

C  =  i  tan  h.  [109! 

If  the  west  end  of  the  axis  is  too  high  (i  positive)  the  telescope  has  to  be  turned 
too  far  west  in  pointing  at  the  star;  the  correction  must  therefore  be  added  to  the 
measured  angle  if  the  mark  is  west  of  the  star,  subtracted  if  east.  If  the  instru- 
ment has  no  striding  level  the  error  must  be  eliminated  as  completely  as  possible 
by  relevelling  between  the  half-sets. 

98.  Diurnal  Aberration. 

Strictly  speaking,  the  computed  azimuth  of  the  star  should  be  corrected  for 
diurnal  aberration,  the  effect  of  which  is  to  make  the  star  appear  farther  east 
than  it  actually  is,  because  the  observer  is  being  carried  due  east  by  the  diurnal 
motion  of  the  earth.  The  correction  is 

f  cos  L  cos  Z 


For  all  but  the  most  precise  observations  it  may  be  taken  as  0^.3  2,  since  the  factor 
—  —  C°S      never  differs  greatly  from  unity. 

COS  h 

*For  the  derivation  of  the  formula  see  Doolittle's  Practical  Astronomy  and 
Hayford's  Geodetic  Astronomy. 


OBSERVATIONS   FOR   AZIMUTH 


159 


Example  i. 
RECORD  OF  AZIMUTH  OBSERVATIONS 

Instrument  (B.  &  B.  No.  3441)  at  South  Meridian  Mark.     Boston,  May  16,  1910. 
(One  division  of  level  =  i5".o.) 


Object. 

3 
tJ 

1 

ex 

e 

"o 

6 
2; 

• 
Chronometer. 

Horizontal  circle. 

Level  readings 
and  angles. 

Vernier  A. 

B. 

W            E 

Polaris  .  . 

nh   24™  358.0 

o°  oo'    oo" 

oo" 

7-o           3-9 

5-8          5-i 

.  27      15.0 

12.8           9.0 

9.0 

28     31-5 

.g 

3-8 

5 

30     oo.o 

Corr.  =  1  2".  5 

Alt.  Polaris  at 

31    20.5 

nh  34™  20*.  5  = 

41°   20'    30" 

32    27.0 

Alt.  Polaris  at 

n*  $jm  04*.  o  = 

Mark... 

6 

*39°  33'    3o" 

30" 

41°  18'  40" 
Mean  horizontal 

angle  = 

66°  35'    35"-  o 

Polaris  .  . 

W             E 

II    42    45.5 

39°  33'    3o" 

30" 

5-i          5-8 

3-3           7-6 

44     09.0 



8.4         13-4 

1 

45     i5-o 

8.4 

1 

46     29.5 

5-o 

Corr.  =  1  6".  5 

47     25-° 

48     54-5' 

Mark... 

6 

*78°  2/    30" 

20" 

Mean  horizontal 

angle  = 

66°  28'    59".  2 

Alt.  Polaris  at 

I2h  OQTO  31*.  5  = 

4i°    15'    40" 

*  Passed  360°. 


l6o  PRACTICAL  ASTRONOMY 

RECORD  OF  TIME  OBSERVATIONS 
Polaris:  —  Chronometer,  12^00™  3is-5;  alt.,  41°  15'  40" 
e  Corvi:  — Chronometer,  12  13  37  .5;  alt.,  25  34  oo 

Polaris:  R.  A.    =     i&  25™    si«.i;         decl.   =  +88°  49'    24".  8 
e  Corvi:    R.  A.   =  i2h     5™    30^.5;         decl.   =  -  22°  07'    21".  o 

Chronometer  R.  A.  Decl. 

a.  Serpentis  (E)       i2h    24™     15*.  7  15^  39™    51*.  6  +6°    42'    20".  7 

e  Hydra:  (W)        12     18      32  .o  8     42       oo  .  5  +6     44     58  .9 

(Lat.  =  42°  21'  oo"  N.;  Long.  =  4^  44™  18*.  o  W.) 

From  these  observations  the  chronometer  is  found  to  be  iow  22s.  i  fast. 

COMPUTATION  OF  AZIMUTH 


Mean 

of  Observed  times 

= 

II* 

37W     25' 

'.6 

Chronometer  correction  =  — 

10            22 

.  i 

Sidereal  time 

=» 

II 

27            03 

•  5 

R.  A.  of  Polaris 

= 

I- 

25            51 

.  i 

Hour  Angle  of  Polaris  = 

10 

01             12 

•  4 

P 

=  150°  18'  06" 

log  cos  L 

= 

Q- 

868670 

log  tan  D 

= 

I. 

687490 

log  cos  L  tan  D 

= 

I. 

556l6o 

cos  L  tan  D 

m 

35- 

9882 

log  sin  Z, 

— 

0- 

82844 

log  cos  P 

"• 

9- 

93884 

log  sin  L  cos  P 

- 

9- 

76728 

sin  L  cos  P 

= 

. 

5852 

denominator 

• 

36. 

5734 

log  sin  P 

•it 

9- 

694985 

log  denom. 

= 

i. 

563165 

log  tan  Z 

= 

8. 

131820 

Z 

= 

0° 

46'    34". 

2 

Curvature  correction  = 

2. 

I 

Azimuth  of  star 

= 

0 

46     32    . 

I 

Measured  angle,  first  half   = 

66° 

35'    35". 

O 

Level  correction 

- 

—  12    . 

5 

Corrected  angle 

• 

66 

35     22  . 

5 

Measured  angle,  second  half  =  66    28     59  .  2 

Level  correction  =  +16  .5 

Corrected  angle  =66    29     15  .  7 

Mark  east  of  star  =  66    32     19  .  i 

Mark  east  of  North  =  65°  45'   47".  o 


OBSERVATIONS   FOR  AZIMUTH  l6l 

Example  2. 

Observed  altitudes  of  Regulus  (east),  Feb.  n,  1908,  in  lat.  42°  21'. 
Altitude  Watch 

17°  05'  ?h  i2m    16* 

17     3i  J4      3i 

17  49  16       07 

18  02  17       20 

The  right  ascension  of  Regulus  is  icA  03™  29s.  i;  the  declination  is  +  12°  24'  57". 
From  these  data  the  sidereal  time  corresponding  to  the  mean  watch  reading 
(7h  I5m  03s.5)  is  found  to  be  4h  53™  42s.  7- 

Observed  horizontal  angles  from  azimuth  mark  to  Polaris. 

(Mark  east  of  north.) 

Telescope  Direct  Time  of  pointing  on  Polaris 

Mark,        o°  oo'  7h  20™    38* 

23       oo 
Third  repetition          201°  48'  23       56 

Mean=  67°  16'.  o  jh  22™  31*.  3 

Telescope  Reversed 

Mark=    o°    oo'  7     27  09 

28  17 

Third  Repetition  201°     54'  29  21 

Mean  =  67°   iS'.o  ?h  28™     15*.  7 

Altitude  of  Polaris  at  jh  20™  38*  =  43°  03' 

Altitude  of  Polaris  at    7     29  21  =  43     01 

Mean  watch  reading  for  Polaris  =  jh    25  m    23*.  5 

Corresponding  sidereal  time  =5     04       04  .  4 

Right  Ascension  of  Polaris  =  i      25       32.3 

Hour-angle  of  Polaris  =3      38       32  .  i 


P  =  4251 
log  p  =  3.  62849 
log  sin  P  =  9.91141 
log  sec  h  =  o.  13611 


log  azimuth  =  3.  67601 
azimuth  =  4743" 

=     i°  19'. o 
Mean  angle   =67     17  .o 

Mark  East  of  North  =  65°  58^.0 

99.   Meridian  by  Polaris  at  Culmination. 

The  following  method  is  given  in  Lalande's  Astronomy  and 
was  practiced  by  Andrew  Ellicott,  in  1785,  on  the  Ohio  and 
Pennsylvania  boundary  survey.  The  direction  of  the  meridian 
is  determined  by  noting  the  instant  when  Polaris  and  some 


162  PRACTICAL  ASTRONOMY 

other  star  are  in  the  same  vertical  plane,  and  then  waiting  a 
certain  interval  of  time,  depending  upon  the  date  and  the  star 
observed,  when  Polaris  will  be  in  the  meridian.  At  this  instant 
Polaris  is  sighted  and  its  direction  then  marked  on  the  ground 
by  means  of  stakes.  The  stars  selected  for  this  observation 
should  be  near  the  hour  circle  through  the  polestar;  that  is, 
their  right  ascensions  should  be  nearly  equal  to  that 
of  the  polestar,  or  else  nearly  i2h  greater.  The  stars 
best  adapted  for  this  purpose  at  the  present  time  are 
8  Cassiopeia  and'£*  Ursa  Majoris. 

The  interval  of  time  between  the  instant  when 
the  star  is  vertically  above  or  beneath  Polaris  and 
}p  the  instant  when  the  latter  is  in  the  meridian  is 
computed  as  follows :  In  Fig.  65  P  is  the  pole,  P'  is 
Polaris,  S  is  the  other  star  (d  Cassiopeia)  and  Z  is 
the  zenith.  At  the  time  when  S  is  vertically  under 
P' ,  ZP'S  is  a  vertical  circle.  The  angle  desired  is 
ZPPf,  the  hour  angle  of  Polaris.  PP'  and  PS,  the 
polar  distances  of  the  stars,  are  known  quantities; 
P'PS  is  the  difference  in  right  ascension,  and  may 
be  obtained  from  the  Ephemeris.  The  triangle  P'PS 
may  therefore  be  solved  for  the  angle  at  P' .  Sub- 
tracting this  from  180°  gives  the  angle  ZP'P\  PPf 
is  known,  and  PZ  is  the  colatitude  of  the  observer. 
The  triangle  ZPfP  may  then  be  solved  for  ZPP',  the  desired 
angle.  Subtracting  ZPP'  from  180°  or  i2h  gives  the  sidereal 
interval  of  time  which  must  elapse  between  the  two 
observations.  The  angle  SPPf  and  the  side  PP'  are  so 
small  that  the  usual  formulae  may  be  simplified,  by  replacing 
sines  by  arcs,  without  appreciably  diminishing  the  accuracy 
of  the  result.  A  similar  solution  may  be  made  for  the  upper 
culmination  of  d  Cassiopeia  or  for  the  two  positions  of  the 
star  f  UrscB  Majoris,  which  is  on  the  opposite  side  of  the 
pole  from  Polaris.  The  above  solution,  using  the  right  ascen- 
sions and  declinations  for  the  date,  gives  the  exact  interval 


OBSERVATIONS    FOR  AZIMUTH  163 

required;  but  for  many  purposes  it  is  sufficient  to  use  a  time 
interval  calculated  from  the  mean  places  of  the  stars  and  for  a 
mean  latitude  of  the  United  States.  The  time  interval  for  the 
star  5  Cassiopeia  for  the  year  1910  is  6m.i  and  for  1918  it  is  about 
nm.2.  For  the  star  f  Ursa  Majoris  the  time  interval  for  the 
year  1910  is  approximately  6m.j,  while  for  1918  it  is  iow.i.  Be- 
ginning with  the  issue  for  1910  the  American  Ephemeris  and 
Nautical  Almanac  gives  values  of  these  intervals,  at  the  end 
of  the  volume,  for  different  latitudes  and  for  different  dates. 
Within  the  limits  of  the  United  States  it  will  generally  be  nec- 
essary to  observe  on  5  Cassiopeia  when  Polaris  is  at  lower 
culmination  and  on  f  Ursa  Majoris  when  Polaris  is  at  upper 
culmination. 

The  determination  of  the  instant  when  the  two  stars  are  in 
the  same  vertical  plane  is  necessarily  approximate,  since  there  is 
some  delay  in  changing  the  telescope  from  one  star  to  the  other. 
The  motion  of  Polaris  is  so  slow,  however,  that  a  very  fair 
degree  of  accuracy  may  be  obtained  by  first  sighting  on  Polaris, 
then  pointing  the  telescope  to  the  altitude  of  the  other  star  (say 
5  Cassiopeia)  and  waiting  until  it  appears  in  the  field;  when 
5  Cassiopeia  is  seen,  sight  again  at  Polaris  to  allow  for  its 
motion  since  the  first  pointing,  turn  the  telescope  again  to 
5  Cassiopeia  and  observe  the  instant  when  it  crosses  the  verti- 
cal hair.  The  motion  of  the  polestar  during  this  short  interval 
may  safely  be  neglected.  The  tabular  interval  of  time  corrected 
to  date  must  be  added  to  the  watch  reading.  When  this  com- 
puted time  arrives,  the  cross  hair  is  to  be  set  accurately  on 
Polaris  and  then  the  telescope  lowered  in  this  vertical  plane  and 
a  mark  set  in  line  with  the  cross  hairs.  The  change  in  the 
azimuth  of  Polaris  in  im  of  time  is  not  far  from  half  a  minute 
of  angle,  so  that  an  error  of  a  few  seconds  in  the  time  of  sighting 
at  Polaris  has  but  little  effect  upon  the  result.  It  is  evident  that 
the  actual  error  of  the  watch  on  local  time  has  no  effect  what- 
ever upon  the  result,  because  the  only  requirement  is  that  the 
interval  should  be  correctly  measured. 


164 


PRACTICAL  ASTRONOMY 


100.   Azimuth  by  Equal  Altitudes  of  a  Star. 

The  meridian  may  be  found  in  a  very  simple  manner  by  means  of  two  equal 
altitudes  of  a  star,  one  east  of  the  meridian  and  one  west.  This  method  has  the 
advantage  that  the  coordinates  of  the  star  are  not  required,  so  that  the  Almanac 
or  other  table  is  not  necessary.  The  method  is  inconvenient  because  it  requires 
two  observations  at  night  several  hours  apart.  It  is  of  special  value  to  surveyors 
in  the  southern  hemisphere,  where  there  is  no  bright  star  near  the  pole.  The  star 
to  be  used  should  be  approaching  the  meridian  (in  the  evening)  and  about  3^  or 
4^  from  it.  The  altitude  should  be  a  convenient  one  for  measuring  with  the  tran- 
sit, and  the  star  should  be  one  that  can  be  identified  with  certainty  6h  or  Sh  later. 
Care  should  be  taken  to  use  a  star  which  will  reach  the  same  altitude  on  the  oppo- 
site side  of  the  meridian  before  daylight  interferes  with  the  observation.  In  the 


northern  hemisphere  one  of  the  stars  in  Cassiopeia  might  be  used.  The  position 
at  the  first  (evening)  observation  would  then  be  at  A  in  Fig.  66.  The  star  should 
be  bisected  with  both  cross  hairs  and  the  altitude  read  and  recorded.  A  note  or 
a  sketch  should  be  made  showing  which  star  is  used.  The  direction  of  the  star 
should  be  marked  on  the  ground,  or  else  the  horizontal  angle  measured  from  some 
reference  mark  to  the  position  of  the  star  at  the  time  of  the  observation.  When 
the  star  is  approaching  the  same  altitude  on  the  opposite  side  of  the  meridian 
(at  B)  the  telescope  should  be  set  at  exactly  the  same  altitude  as  was  read  at  the 
first  observation.  When  the  star  comes  into  the  field  it  is  bisected  with  the  ver- 
tical cross  hair  and  followed  in  azimuth  until  it  reaches  the  horizontal  hair.  The 
motion  in  azimuth  should  be  stopped  at  this  instant.  Another  point  is  then  set 
on  the  ground  (at  same  distance  from  the  transit  as  the  first)  or  else  another  angle 


OBSERVATIONS   FOR  AZIMUTH  165 

is  turned  to  the  same  reference  mark.  The  bisector  of  the  angle  between  the  two 
directions  is  the  meridian  line  through  the  transit  point.  It  is  evident  that  the 
index  and  refraction  errors  are  eliminated,  because  they  are  alike  for  the  two 
observations.  If  one  observation  is  made  with  the  telescope  direct  and  the  other 
with  the  telescope  reversed,  the  other  instrumental  errors  will  be  eliminated.  Care 
should  be  taken  to  level  the  instrument  just  before  the  observations.  The  accu- 
racy of  the  final  result  may  be  increased  by  observing  the  star  at  several  different 
altitudes  and  using  the  mean  value  of  the  different  results. 

101.  Observation  for  Meridian  by  Equal  Altitudes  of  the  Sun  in  the  Forenoon 
and  in  the  Afternoon. 

This  observation  consists  in  measuring  the  horizontal  angle  between  the  mark 
and  the  sun  when  it  has  a  certain  altitude  in  the  forenoon  and  measuring  the 
angle  again  to  the  sun  when  it  has  an  equal  altitude  in  the  afternoon.  Since  the 
sun's  declination  will  change  during  the  interval,  the  mean  of  the  two  angles  will 
not  be  the  true  angle  between  the  meridian  and  the  mark,  but  will  require  a  small 
correction.  The  angle  between  the  south  point  of  the  meridian  and  the  point 
midway  between  the  two  directions  of  the  sun  is  given  by  the  equation 

Correction  =  .,  [„,] 


in  which  d  is  the  hourly  change  in  declination  multiplied  by  the  number  of  hours 
elapsed  between  the  two  observations,  L  is  the  latitude,  and  P  is  the  hour  angle 
of  the  sun,  or  approximately  half  the  elapsed  interval  of  time.  The  correction 
depends  upon  the  change  in  the  declination,  not  upon  its  absolute  value,  so  that 
the  hourly  change  may  be  taken  with  sufficient  accuracy  from  the  Almanac  for 
any  year  for  the  corresponding  date. 

In  making  the  observation  the  instrument  is  set  up  at  one  end  of  the  line  whose 
azimuth  is  to  be  determined,  and  the  plate  vernier  set  at  o°.  The  vertical  cross 
hair  is  set  on  the  mark  and  the  lower  clamp  tightened.  The  sun  glass  is  then  put 
in  position,  the  upper  clamp  loosened,  and  the  telescope  pointed  at  the  sun. 
It  is  not  necessary  to  observe  on  both  edges  of  the  sun,  but  is  sufficient  to  sight, 
say,  the  lower  limb  at  both  observations,  and  to  sight  the  vertical  cross  hair  on 
the  opposite  limb  in  the  afternoon  from  that  used  in  the  forenoon.  The  hori- 
zontal hair  is  therefore  set  on  the  lower  limb  and  the  vertical  cross  hair  on  the  left 
limb.  When  the  instrument  is  in  this  position  the  time  should  be  noted  as  accurately 
as  possible.  The  altitude  and  the  horizontal  angle  are  both  read.  In  the  after- 
noon the  instrument  is  set  up  at  the  same  point,  and  the  same  observation  is  made, 
except  that  the  vertical  hair  is  now  sighted  on  the  right  limb;  the  horizontal  hair 
is  set  on  the  lower  limb  as  before.  A  few  minutes  before  the  sun  reaches  an  alti- 
tude equal  to  that  observed  in  the  morning  the  vertical  arc  is  set  to  read  exactly 
the  same  altitude  as  was  read  at  the  first  observation.  As  the  sun's  altitude  de- 
creases the  vertical  hair  is  kept  tangent  to  the  right  limb  until  the  lower  edge 
of  the  sun  is  in  contact  with  the  horizontal  hair.  At  this  instant  the  time  is  again 
noted  accurately;  the  horizontal  angle  is  then  read.  The  mean  of  the  two  circle 
readings,  corrected  for  the  effect  of  change  in  declination,  is  the  angle  from  the 


1 66  PRACTICAL  ASTRONOMY 

mark  to  the  south  point  of  the  horizon.     The  algebraic  sign  of  the  correction  is 
determined  from  the  fact  that  if  the  sun  is  going  north  the  mean  of  the  two  ver- 
nier readings  lies  to  the  west  of  the  south  point,  and  vice  versa.     The  precision 
of  the  result  may  be   increased  by  taking  several  forenoon  observations  in  suc- 
cession and   corresponding  observations  in  the  afternoon. 
Example. 
Lat.  42°  18'  N.     Apr.  19,  1906. 

A.M.  Observations.  P.M.  Observations. 

Reading  on  Mark,  o°  oo'  oo"  Reading  on  Mark,  o°  oo'  oo" 

(  Alt.,  24°  58'  (  Alt.,  24°  58' 

U  &  L  limbs  ]  Hor.  Angle,  357°  14'  15"  U  &  R  limbs  ]  Hor.  angle,  162°  2S'oo" 

(  Time,  7^  igm  30*  (  Time  4^  i2m  i$s 

%  elapsed  time  =  4h  26™  22s 

P  =  66°  35'  30"  Incr.  in  decl.  =  +52"  X  4h.  44 

log  sin  P  =  9.96270  =  +  230".  9 

log  cos  L  =  9.86902 

9.83172  Mean  Circle  Reading    =  79°  51'  08" 

log  230".  9  =  2.36342  Correction  =          5  40 


2.53170  True  Angle  =  S  79°  45'    28"  E. 

Corr.  =  340". 2  Azimuth  =    280°  14'  32" 

102.   Azimuth  of  Sun  near  Noon. 

The  azimuth  of  the  sun  near  noon  may  be  determined  by  means  of  Equa.  [30], 
provided  the  local  apparent  time  is  known  or  can  be  computed.  If  the  longitude 
and  the  watch  correction  on  Standard  Time  are  known  within  one  or  two  seconds 
the  local  apparent  time  may  be  readily  calculated.  This  method  may  be  useful 
when  it  is  desired  to  obtain  a  meridian  during  the  middle  of  the  day,  because  the 
other  methods  are  not  then  applicable. 

If,  for  example,  an  observation  has  been  made  in  the  forenoon  from  which  a 
reliable  watch  correction  may  be  computed,  then  this  correction  may  be  used  in 
the  azimuth  computation  for  the  observation  near  noon;  or  if  the  Standard  Time 
can  be  obtained  accurately  by  a  comparison  at  noon  and  the  longitude  can  be 
obtained  from  a  map  within  about  1000  feet,  the  local  apparent  time  may  be 
found  with  sufficient  accuracy.  This  method  is  not  usually  convenient  in  mid- 
summer, on  account  of  the  high  altitude  of  the  sun,  but  if  the  altitude  is  not 
greater  than  about  50°  the  method  may  be  used  without  difficulty.  The  obser- 
vations are  made  exactly  as  in  Art.  93,  except  that  the  time  of  each  pointing  is 
determined  more  precisely;  the  accuracy  of  the  result  depends  very  largely  upon 
the  accuracy  with  which  the  hour  angle  of  the  sun  can  be  computed,  and  great 
care  must  therefore  be  used  in  determining  the  time.  The  observed  watch  read- 
ing is  corrected  for  the  known  error  of  the  watch,  and  is  then  converted  into  local 
apparent  time.  The  local  apparent  time  converted  into  degrees  is  the  angle  at 
the  pole,  P.  The  azimuth  is  then  found  by  the  formula 

sin  Z  =  sin  P  sec  h  cos  D.  [30] 

Errors  in  the  time  and  the  longitude  produce  large  errors  in  Z,  so  this  method 
should  not  be  used  unless  both  can  be  determined  with  certainty. 


OBSERVATIONS  FOR  AZIMUTH 


167 


Example. 

Observation  on  the  sun  for  azimuth. 

Lat.  42°  21'.    Long.  4h  44™  i8s  W.     Date,  Feb.  5,  1910. 


Hor.  Circle. 

Mark,    o°  oo' 

app.  L  &  L  limbs,      29     01 

app.  U  &  R  limbs,     28     39 

Mean,  28°  50' 


Refr., 


Vert.  Circle. 

3i°  49' 
31    16 

3i°  32'.  5 

i  .  6    Watch  corr. 


Watch. 
(30*  fast) 

11/143™  22* 

1  1    44     20 


h  =  31°  3<>'- 9 
D  =  -  1 6°  06'   04".  5 


E.  S.T.    = 


45".  17X4*.  7       = 


3     32   -3 


D  =  —  1 6°  02'   32".  2 

Eq.  t.    =  i4TO  08*.  05 
.217  X4h.  7  I  -02 


Eq.  t. 


07 


L.  M.  T. 

Eq.  t. 

L.  A.  T. 
P 


log  sin  P 
log  cos  D 
log  sec  h 

log  sin  Z 
Hor.  Circle 


30 


15     42 


14  09  .  I 

nA44m  53s.  9, 

15  m  o6s.  i 
3°   46'.  5 


8.  81847 
9.98275 
o.  06930 


8.87052 

4°     15'- 4 

28     50 


Azimuth  =  S  33°  05'.  4  E 
=  326°  54'.  6 

103.    Approximate  Azimuth  of  Polaris  when  the  Time  is  Known. 

If  the  error  of  the  watch  is  known  within  a  half  minute  or  so,  the  azimuth  of 
Polaris  may  be  computed  to  the  nearest  minute,  i.e.,  with  sufficient  precision  for 
the  purpose  of  checking  the  angles  of  a  transit  survey.  The  horizontal  angle  be- 
tween Polaris  and  a  reference  mark  should  be  measured  and  the  watch  time  of  the 
pointing  on  the  star  noted.  It  is  desirable  to  measure  also  the  altitude  of  Polaris 
at  the  instant,  although  this  is  not  absolutely  necessary.  A  convenient  time  to 
make  this  observation  is  just  before  dark,  when  both  Polaris  and  the  crosshairs  can 
be  seen  without  using  artificial  light. 

The  formula  employed  in  computing  the  azimuth  is 

Z'  =  p'  sin  P  sec  h.  [106] 

The  hour  angle  of  Polaris  (P)  at  the  time  of  the  observation  may  be  found  with 
the  aid  of  Table  V,  p.  188.  The  observed  watch  time  should  be  corrected  for  any 
known  error  and  then  converted  into  local  mean  astronomical  time.  The  hour 
angle  is  then  found  by  subtracting  from  this  result  the  time  of  the  next  preceding 
upper  culmination  of  Polaris,  as  found  in  Table  V  for  the  day  and  year.  If  the 
local  time  is  less  than  the  time  of  culmination,  it  should  be  increased  by  24^  before 


1 68  PRACTICAL  ASTRONOMY 

subtracting.  The  resulting  interval  of  time  should  now  be  converted  into  sidereal 
units  by  adding  ios  for  every  hour  in  the  interval.  (This  correction  may  be  taken 
from  Table  III.)  This  result  is  the  hour  angle  P  and  shows  the  position  of  Polaris 
at  the  instant  of  the  observation. 

It  is  important  that  the  time  of  culmination  employed  should  be  that  preceding 
the  time  of  the  observation.  Note  that  if  the  time  of  culmination  is  greater 
than  the  observed  time  the  preceding  culmination  occurs  on  the  preceding  astro- 
nomical date  (as  in  the  example  below). 

To  obtain  the  product  p'  sin  P  required  by  the  formula  enter  Table  E  with  the 
hour  angle  P  and  interpolate  for  the  year  of  the  observation.  If  the  number  of 
hours  in  P  is  greater  than  6/l  and  less  than  i2h,  subtract  from  i2h  and  use  the  re- 
mainder; if  greater  than  i2h  and  less  than  i8A,  subtract  i2h;  if  greater  than  iSh, 
subtract  from  24^,  and  enter  Table  E  with  the  difference. 

To  obtain  the  azimuth  p'  sin  P  must  be  multiplied  by  sec  //,  which  may  be  found 
in  Table  F.*  The  result  is  the  azimuth  of  Polaris  at  the  time  of  the  observation. 
Observe  that  if  the  hour  angle  is  less  than  i2h  Polaris  is  west  of  the  meridian;  if 
greater  than  i2h,  it  is  east. 

Example. 

On  May  8,  1916,  in  lat.  40°  N.,  long.  71°  W.,  the  angle  was  measured  between 
olaris  and  a  reference  mark,  the  observation  on  Pol 
Eastern  time;  watch  was  im  fast;  altitude,  39°  06'. 


Obs'd  time        7*    45™  In  Table  V,  May  i,        22*  50^.0 

Error                     —  i  6d                      23   . 5 

7A     44m  1915,  May  7,t      22*  26™.  5 

Long.  corr.                i6w  —2.3 

Loc.  M.  T.        8*     oom  (P.M.)  Time  of  U.C.  1916           22*  24™.  2 
32*  oo™ 

Time  of  U.  C.    22     24.  2  From  Table  E,  opp.  2h  22™.  6,  p'  sin  P  =  40'.  o 

ios  X  ^|A                      i  .6  40'.  o  X  sec  39°  06'  =  51'.  6 

H.  A.                    9*    37TO.  4  .'.  The  azimuth  is  o°  51'.  6  West. 

(I2A  -    2h   22m.6) 


*  The  altitude  to  be  used  is  that  measured  at  the  time  of  the  observation;  but 
if  this  measurement  has  been  omitted,  the  altitude  may  be  estimated  closely,  if  the 
latitude  is  known,  by  noting  the  position  of  the  star  with  reference  to  the  pole,  and 
remembering  that  the  altitude  of  the  pole  is  equal  to  the  latitude  of  the  place. 
See  Art.  62,  p.  99,  and  star  map  I. 

t  The  preceding  culmination. 


OBSERVATIONS  FOR  AZIMUTH 


i68a 


TABLE  E 
Values  of  p'  sin  P  for  Polaris  (in  minutes)  $ 


H.A. 

1910 

1920 

1930 

H.A. 

1910 

1920 

1930 

H.A. 

1910 

1920 

1930 

H.A. 

1910 

1920 

1930 

m 
O 

0.0 

o.o 

o.o 

h  m 
I  00 

18.2 

17-4 

16.6 

h  m 
200 

35-2 

33-7 

32.1 

h  m 

400 

61.0 

58.3 

55.6 

4 

1.2 

1.2 

I.I 

04 

19.4 

18.6 

17-7 

08 

37-3 

35-7 

34.1 

08 

62.2 

59-S 

56  7 

8 

2.5 

2-4 

2.2 

08 

20.6 

19-7 

18.8 

16 

39-4 

37-7 

35-9 

16 

63.3 

60.5 

57.8 

12 

3-7 

3-5 

3-4 

12 

21.8 

20.8 

19.9 

24 

41.4 

39-6 

37-8 

24 

64-4 

61.5 

58.7 

16 

4-9 

4-7 

4-5 

16 

22.9 

21.9 

20.9 

32 

43-4 

41-5 

39-6 

32 

65.3 

62.4 

59-6 

20 

6.1 

5-9 

5-6 

20 

24.1 

23.0 

22.  0 

40 

45-3 

43-3 

41-3 

40 

66.2 

63-3 

60.4 

24 

7-4 

7.0 

6.7 

24 

25.2 

24.1 

23-0 

48 

47-1 

45-1 

43-0 

48 

67.0 

64.1 

61.1 

28 

8.6 

8.2 

7.8 

28 

26.4 

25.2 

24.1 

56 

48.9 

46.8 

44-6 

56 

67-7 

64.7 

61.8 

32 

9-8 

9-4 

8.9 

32 

27-5 

26.3 

25.1 

304 

50.7 

48.4 

46.2 

504 

68.3 

65.3 

62.3 

36 

II.  O 

10.5 

IO.I 

36 

28.6 

27-4 

26.  1 

12 

52-4 

50.0 

47-7 

12 

68.9 

65.9 

62.9 

40 

12.2 

II.  7 

II.  2 

40 

29.8 

28.5 

27.2 

2O 

54-0 

51-6 

49-2 

2O 

69.4 

66.3 

63-3 

44 

13-4 

12.8 

12.3 

44 

30.9 

29-5 

28.2 

28 

55-5 

53-1 

50.6 

28 

69.8 

66.7 

63.6 

48 

14-6 

14.0 

13-4 

48 

32.0 

30.6 

29.2 

36 

57.0 

54-5 

52.0 

36 

70.1 

67.0 

63.9 

52 

15.8 

I5-I 

14-4 

52 

33-1 

31-6 

30.2 

44 

58.4 

55-8 

53-3 

44 

70.3 

67.2 

64.1 

56 

17.0 

16.3 

15-5 

56 

34-2 

32.7 

3L2 

52 

59-7 

57-1 

54-5 

52 

70.4 

67.3 

64.2 

60 

18.2 

17-4 

16.6 

2  OO 

35-2 

33-7 

32.1 

400 

61.0 

58.3 

55-6 

6  oo 

70.4 

67-4 

64.3 

TABLE  P.  — NATURAL  SECANTS 


10° 

.015 

20° 

.064 

3C° 

.155 

40° 

.305 

50° 

-556 

II 

.019 

21 

.071 

31 

.  167 

41 

-325 

Si 

.589 

12 

.022 

22 

.079 

32 

•  179 

42 

.346 

52 

.624 

13 

.026 

23 

.086 

33 

.192 

43 

.367 

53 

.661 

14 

.031 

24 

.095 

34 

.206 

44 

•  390 

54 

.701 

15 

035 

25 

.103 

35 

.221 

45 

.414 

55 

•  743 

16 

.040 

26 

.113 

36 

.236 

46 

.440 

56 

.788 

17 

.046 

27 

.122 

37 

.252 

47 

.466 

57 

.836 

18 

-051 

28 

.133 

38 

.269 

48 

•  494 

58 

.887 

19 

.058 

29 

•  143 

39 

.287 

49 

.524 

59 

•  942 

Combining  the  Preceding  Methods  of  Observation. 

From  the  foregoing  descriptions  of  field  methods  of  observing,  it  will  be  seen  that 
but  few  of  these  methods  are  quite  independent  of  the  data  obtained  by  other  obser- 
vations; in  the  practice  of  the  engineer  it  often  happens  that  no  one  of  the  quanti- 
ties which  he  desires  can  be  completely  determined  until  some  or  all  of  the  others 
are  known  approximately.  The  latitude  may  be  determined  directly  by  observing 
a  star  at  culmination,  but  it  may  be  inconvenient  or  impracticable  to  wait  until 
Polaris  or  a  southern  star  crosses  the  meridian.  In  all  of  the  methods  of  determin- 
ing time  it  is  necessary  to  know  either  the  latitude  or  the  direction  of  the  meridian 
before  the  time  can  be  computed.  Furthermore,  in  all  of  the  methods  of  deter- 
mining azimuth  either  the  time,  the  latitude,  or  both  must  be  known.  Where  all 
of  these  quantities  are  entangled  it  is  necessary  to  arrive  at  the  true  values  by  a 
series  of  approximations.  In  most  cases,  however,  very  few  approximations  are 
necessary  to  give  the  greatest  accuracy  afforded  by  the  observations. 

If  it  is  necessary  to  determine  a  precise  azimuth,  and  nothing  whatever  is  known 
in  regard  to  the  local  time  or  the  latitude  of  the  point,  then  all  three  may  be  accu- 
rately determined  by  making  observations  of  transits  across  the  vertical  circle 


l68b  PRACTICAL  ASTRONOMY 

through  Polaris,  measuring  the  altitudes  of  all  the  stars,  and  then  repeating  the 
horizontal  angle  between  Polaris  and  an  azimuth  mark.  The  measured  altitudes 
of  Polaris  and  the  time-star  make  it  possible  to  compute  the  latitude  with  sufficient 
precision  for  determining  the  time.  When  the  time  is  known  a  more  precise  latitude 
may  be  found  if  desired.  If  the  instrument  has  a  vertical  arc  of  only  180°,  then  the 
altitudes  of  the  southern  stars  may  be  measured  and  the  first  approximation  to  the 
latitude  found  from  these  observations.  The  altitudes  of  Polaris  may  then  be 
calculated  closely  enough  for  computing  Equa.  [106].  After  the  time  and  the  lati- 
tude are  known  the  azimuth  is  found  directly.  By  using  the  instrument  in  the  two 
positions  and  increasing  the  number  of  observations  the  precision  of  all  of  the  re- 
sults may  be  increased. 

In  a  similar  manner  the  method  of  equal  altitudes  (Arts.  80-82)  may  be  combined 
with  measures  of  the  altitude  of  the  polestar  and  observations  for  azimuth.  By 
selecting  a  pair  of  stars  having  a  large  difference  in  right  ascension  or  a  small  differ- 
ence in  declination,  the  time  may  be  fairly  well  determined  by  using  an  estimated 
latitude  obtained  by  estimating  a  correction  to  the  observed  altitude  of  Polaris. 
When  the  time  is  known  approximately,  a  new  value  of  the  latitude  may  be  obtained, 
and  with  this  new  latitude  the  time  may  be  recomputed.  The  azimuth  may  then  be 
found  as  before. 

A  very  rapid  but  not  very  precise  way  of  determining  these  three  quantities 
and  also  checking  the  azimuth  is  to  sight  first  on  the  mark,  then  to  sight  on  the  pole- 
star,  reading  both  the  horizontal  and  vertical  angles,  and  finally  to  sight  on  a  prime 
vertical  star,  reading  both  angles.  Using  an  estimated  latitude,  the  PZS  triangle 
may  be  solved  for  P;  with  this  value  of  P  a  closer  value  of  the  latitude  is  found, 
and  the  hour  angle  is  then  recomputed.  Now  that  the  latitude  and  the  time  are 
known  the  azimuth  may  be  determined  from  the  polestar  and  checked  by  the 
azimuth  from  the  star  near  the  prime  vertical. 

It  is  well  when  determining  azimuths  for  surveying  purposes  to  obtain  checks 
by  methods  which  are  independent  of  one  another.  For  example,  if  the  azimuth 
is  being  found  by  angles  measured  to  Polaris,  a  check  may  be  obtained  by  turning 
an  angle  from  some  star  near  the  prime  vertical  (Art.  77)  and  measuring  its  alti- 
tude simultaneously.  Observations  made  on  both  east  and  west  stars  will  increase 
the  accuracy.  The  azimuth  thus  computed  is  inferior  in  accuracy  to  that  found 
from  Polaris,  but  the  fact  that  it  is  independent  makes  it  a  valuable  check  against 
mistakes  or  large  errors  in  the  Polaris  observations.  A  sun  observation  made  late 
in  the  afternoon  may  be  used  in  a  similar  way  to  check  an  evening  observation 
on  Polaris. 

Questions  and  Problems 

1.  What  error  is  caused  by  making  the  approximations  in  deriving  formula 

[101]? 

2.  Derive  formula  [106]. 

3.  Show  that  if  the  declination  is  less  than  the  latitude  the  most  favorable 
conditions  for  determining  azimuth  by  an  altitude  of  the  sun  occur  when  the  sun 


OBSERVATIONS   FOR  AZIMUTH  169 

is  between  the  six-hour  circle  and  the  prime  vertical.     For  greatest  accuracy 

A7  A7 

—  and  -jj  should  be  a  minimum.     Differentiate  Equa.  [13],  p.  32,  and  simplify 
by  means  of  [14]  and  [15]. 

4.   Show  that  the  factor  cos  L  cos  Z  sec  h  (Equa.  [no],  p.  158)  is  always  nearly 
equal  to  unity. 

5.  Compute  the  approximate  local  mean  time  of  eastern  elongation  of  Polaris 
on  Sept.  10.     R.  A.  of  Polaris,  ih  25™.     See  Art.  63,  p.  101,  for  an  approximate 
method  of  finding  the  R.  A.  of  the  mean  sun.     Use  5^  55™  for  the  hour  angle  of 
Polaris  at  elongation  (see  Art.  91,  p.  14?)- 

6.  Observation  on  sun  May  15,  1906,  for  azimuth.     Vernier  A,  on  mark,  read 
o°  oo'.     On  the  sun,  right  and  lower  limbs,  vertical  circle  read  43°  36';  vernier  A 
read  168°  59'  (right-handed);    E.  S.  T.,  2^52^458  P.M.     Upper  and  left   limbs, 
vernier  A  read  168°  52';  vertical  circle  read  42°  33';  E-  s-  T->  2/l  55m  37s  P.M.  Dec- 
lination at  G.  M.  N.  =  +  18°  42'  43"-  6;  diff.  for  i*  =  +  35".94.     The  latitude 
of  the  place  is  42°  17'  N.;  longitude  71°  05'  W.     Compute  the  azimuth  of  the  mark. 
The  equation  of  time  at  G.  M.  N.  is  +  3W  503.96 ;  diff.  for  ih  is  —  o'.ooi. 

7.  Compute  the  azimuth  of  Jupiter  from  the  data  given  in  Art.  77,  p.  124. 

8.  Prove  that  the  horizontal  angle  between  the  centre  of  the  sun  and  the 
right  or  left  limb  is  S  sec  h  where  S  is  the  apparent  angular  semidiameter  and  h  is 
the  apparent  altitude. 

9.  Prove  that  the  level  correction  (Art.  97)  is  i  tan  h. 

10.  Why  could  not  Equa.  [106],  p.  157,  be  used  in  place  of  Equa.  [30],  p.  35, 
in  the  method  of  Art.  102  ? 

n.  If  there  is  an  error  of  4*  in  the  assumed  value  of  the  watch  correction  and 
an  azimuth  is  determined  by  the  method  of  Art.  102  (near  noon),  what  would  be 
the  relative  effect  of  this  error  when  the  sun  is  on  the  equator  and  when  it  is  23° 
South  ?  Assume  the  latitude  to  be  45°  N.  (See  Table  B,  p.  88.) 

12.  Make  a  set  of  azimuth  observations  by  the  method  of  Art.  93  (three  point- 
ings in  each  position  of  the  instrument),  and  plot  a  curve  using  altitudes  for  ordinates 
and  horizontal  angles  for  abscissae;  also  plot  a  curve  using  altitudes  and  times  for 
the  two  coordinates. 


CHAPTER  XIV 
NAUTICAL  ASTRONOMY 

104.  Observations  at  Sea. 

The  problems  of  determining  a  ship's  position  at  sea  and  the 
bearing  of  a  celestial  object  at  any  time  are  based  upon  exactly 
the  same  principles  as  the  surveyor's  problems  of  determining 
his  position  on  land  and  the  azimuth  of  a  line  of  a  survey.  The 
method  of  making  the  observations,  however,  is  different, 
since  the  use  of  instruments  requiring  a  stable  support,  such  as 
the  transit  and  the  artificial  horizon,  is  not  practicable  at  sea. 
The  sextant  does  not  require  a  stable  support  and  is  well  adapted 
to  making  observations  at  sea.  Since  the  sextant  can  be  used 
only  to  measure  the  angle  between  two  visible  points,  it  is 
necessary  to  measure  all  altitudes  from  the  sea-horizon  and  to 
make  the  proper  correction  for  dip. 

Determination  of  Latitude  at  Sea 

105.  Latitude  by  Noon  Altitude  of  Sun. 

The  determination  of  latitude  by  measuring  the  maximum 
altitude  of  the  sun's  lower  limb  at  noon  is  made  in  exactly  the 
same  way  as  described  in  Art.  66.  The  observation  should  be 
begun  a  little  before  local  apparent  noon  and  altitudes  measured 
in  quick  succession  until  the  maximum  is  reached.  In  measur- 
ing an  altitude  above  the  sea-horizon  the  observer  should  bring 
the  sun's  image  down  until  the  lower  limb  appears  to  be  in 
contact  with  the  horizon  line.  The  sextant  should  then  be 
tipped  by  rotating  right  and  left  about  the  axis  of  the  telescope 
so  as  to  make  the  sun's  image  describe  an  arc;  if  the  lower  limb 
of  the  sun  drops  below  the  horizon  at  any  point,  the  measured 
altitude  is  too  great,  and  the  index  arm  should  be  moved  until 
the  sun's  image  is  just  tangent  to  the  horizon  when  at  the  lowest 

170 


NAUTICAL  ASTRONOMY 


171 


point  of  the  arc.     (Fig.  67.)     This  method  is  illustrated  by  the 
following  example. 

Example. 

Observed  altitude  of  sun's  lower  limb  69°  21'  30",  bearing  north.  Index  cor- 
rection =  —  i'  10";  height  of  eye  =  18  feet;  sun's  declination  at  G.  A.  N.  = 
N  8°  59'  32";  diff.  ih  =  -f-  54/'.43.  Approx.  lat.  =  11°  30'  S;  approx.  long.  = 
i  A  oo™  W. 


Obs'd  alt.  =  69°  21'  30" 
Corr.  =      +10    22 


Alt.  centre  =  69°  31'  52" 
Declination  =    9    oo    26 


Colatitude  =  78°  32'  18" 
Latitude  =  11°  27'  42"  S 


Corrections 

I.  C.  =  -    i'  10" 

Dip  =  —    4  09 

r  &  p  =  —    o  17 

S.  D.  =  +  15  58 

Corr.  =  +  10'  22" 


Decl  +  8°  59'  32" 

+  54 


9°  oo'  26" 


O 


Horizon 


FIG.  67 


1 06.   Latitude  by  Ex- Meridian  Altitudes. 

If  for  any  reason  the  noon  altitude  has  been  lost,  an  altitude  may  be  measured 
near  noon  and  this  altitude  corrected  to  the  corresponding  noon  altitude  by 
Equa.  [72].  In  order  to  make  this  "  reduction  to  the  meridian  "  it  is  necessary 
to  know  the  sun's  hour  angle.  If  the  altitude  is  taken  within  a  few  minutes  of 
noon  the  reduction  may  be  made  by  the  more  convenient  formula,  [74];  in  practice 
this  is  done  by  means  of  tables. 

Example. 

Observed  altitude  Q  Jan.  20,  1910  =  20°  05';  I.  C.  =  o;  G.  A.  T.  =  i^  35"* 
28s;  lat.  by  dead  reckoning  =  49°  20'  N;  longitude  by  dead  reckoning  =  i^  05"* 
20*;  height  of  eye  =  16  feet;  de'cl.  at  G.  M.  N.  =  20°  15'  02"  S;  diff.  for  i^  = 
+  32".o;  S.D.  =  16'  17". 


Decl 
Diff. 

G.  A.  T.  =  i*   35™ 
Long.  W  =  i     05 

28*            S.D.  =  +  16'      17" 

20                     I.  C.     =                         00 

cos  L 
cos  D 
vers  P 

Corr. 
sin  h 

•  sin  hm 

Dec?. 
Colat. 

=  9.  8140 
=  9-9723 

H.  A.  =  o71  30™ 

—     *7°      f)' 

-Dip.                 3    55 
o8s            r&p   =  —      2     27 

7.7224 
=  -0053 

7    32 

.  G.  M.  N.  =  20°  15' 
for  i  h.  i       = 

Corr.    =  +      9'  55" 
Obs.  Alt.  =  20°  05 

=  -3514 

=  20°  34' 
=  2O  14 
=  40°  49' 

35       True  Alt.  =  20°   14'  55" 

Decl.  =  20°  14' 

27"  S 

Lat.  =  49°  ii' 


172  PRACTICAL  ASTRONOMY 

Determination  of  Longitude  at  Sea 

107.  By  the  Greenwich  Time  and  the  Sun's  Altitude. 

The  usual  method  of  finding  the  longitude  at  sea  is  to  determine 
the  local  mean  time  from  an  observed  altitude  of  the  sun  (Art. 
76)  and  to  compare  this-  with  the  Greenwich  Mean  Time  as 
shown  by  the  chronometer.  The  error  of  the  chronometer  at 
some  previous  date  and  its  daily  gain  or  loss  are  supposed  to  be 
known.  This  is  the  same  in  principle  as  the  method  of  Art.  86. 
The  value  of  the  latitude  used  in  solving  the  PZS  triangle  must 
be  that  of  the  ship  at  the  time  the  observation  is  made;  this 
latitude  must  be  found  by  correcting  the  latitude  by  observation 
at  the  previous  noon  for  the  run  of  the  ship  in  the  interval. 
This  is  called  the  latitude  by  "  dead  reckoning."  On  account 
of  the  large  errors  which  may  enter  into  this  estimated  latitude 
it  is  important  that  the  observation  ("  time-sight  ")  should  be 
made  when  the  sun  is  near  the  prime  vertical. 

Example. 

True  alt.  0  May  19,  1910  (P.M.)  =  44°  05';  G.  M.  T.  =  6h  55™  io».  Lat.  by 
dead  reckoning  =  42°  oo'  N;  decl.  at  G.  M.  N.  =  19°  38'  20"  N;  diff.  i*  = 
+  32". 7;  equa.  of  time  =  —  3m  44s. i;  deer,  per  ih  =  os.i. 

L  =  42°  oo'  sec  =  .  1289     Decl.  G.  M.  N.    19°  38'   20"  N 
D  =  19   42    sec  =  .0262  +32".;  X  6^.9   =  3    46 

cos    =  .9252      L  —  D  =  22    18  Cor'd.  decl.  =  19°  42'   06"  N 

sin    =  .  6957  h  =  44   05 

diff.  =  .  2295  log    =  9.  3608 

logvers  =  9.5159          Equa.G.M.N.=  —  3m44s- 1 
H.  A.  =  3A  nmo7*        os.iX6A.9  =  .7 

Eq.  t.  =  —    3    43 

Cor'd  Eq.  t.    =  -  3™  43s-  4 

L.  M.  T.  =  3*  07™  24* 
G.  M.  T.  =  6     55     10 

Long.  W  =  3^  47m  46* 
=  56°  56' *W. 

1 08.  By  a  Lunar  Distance. 

The  accuracy  of  the  preceding  method  is  wholly  dependent 
upon  the  accuracy  of  the  chronometer  giving  the  Greenwich 
time.  With  steam  vessels  making  short  trips  and  carrying 


NAUTICAL  ASTRONOMY  173 

several  chronometers  this  method  gives  the  longitude  with 
sufficient  accuracy.  In  the  days  when  commerce  was  carried 
on  chiefly  by  means  of  sailing  vessels  the  voyages  were  of  long 
duration,  and  consequently  the  error  of  the  chronometer  could 
be  verified  only  at  long  intervals;  furthermore,  the  chronom- 
eters of  that  time  were  far  less  perfect  than  those  of  to-day,  and 
their  rates  were  subject  to  greater  irregularities.  Under  these 
circumstances  the  method  just  described  sometimes  became 
wholly  unreliable;  in  such  cases  the 
method  of  "  lunar  distance "  was 
used.  Although  this  method  is 
necessarily  of  inferior  accuracy  it  has 
the  advantage  of  being  entirely  inde- 
pendent of  the  chronometer  time.  In 
the  Nautical  Almanac  previous  to  the 
issue  for  1912  there  were  given  the 
geocentric  distances  of  the  moon 
from  several  bright  stars,  planets,  and  FlG  68 

the  sun,  for  every  3^  of  Greenwich 

Mean  Time.  If  a  lunar  distance  were  measured  at  sea  and  this 
distance  reduced  to  the  centre  of  the  earth,  the  corresponding 
instant  of  G.  M.  T.  could  be  found  by  interpolation  in  these 
tables. 

The  observation  requires  that  the  altitudes  of  the  moon  and 
the  sun  or  star  should  be  measured  simultaneously  with  the 
distance,  and  that  the  chronometer  should  be  read  at  the  same 
instant.  In  Fig.  68  let  Z  be  the  observer's  zenith,  Mr  the  appar- 
ent and  M  the  true  position  of  the  moon,  and  S'  and  S  the  appar- 
ent and  true  positions  of  the  sun.  The  sun's  apparent  position 
is  higher  than  its  true  position  because  its  refraction  is  greater 
than  its  parallax.  The  moon's  true  position  is  higher  than  its 
apparent  position  because  the  parallax  correction  is  the  greater. 
The  measured  distance  S'M'  is  to  be  reduced  to  the  true  dis- 
tance SM .  In  the  triangle  ZS'M'  the  three  sides  have  been 
measured  and  the  angle  Z  may  be  computed.  Then  in  the 


174  PRACTICAL  ASTRONOMY 

triangle  ZSM  the  angle  Z  and  the  sides  ZS  and  ZM  are  known, 
because  the  refraction  and  parallax  corrections  are  known,  and 
MS  may  be  computed.  By  interpolating  in  the  tables,  the 
true  G.  M.  T.  corresponding  to  the  instant  of  this  observation 
may  be  obtained,  the  difference  between  this  and  the  observed 
chronometer  time  being  the  error  of  the  chronometer  on  G.  M.T. 
The  longitude  may  then  be  found  by  comparing  the  true  G.  M.  T. 
with  the  local  time  computed  from  the  sun's  altitude. 

In  the  Ephemeris  for  1912  the  tables  of  lunar  distances  have 
been  omitted,  as  lunar  observations  are  no  longer  considered 
to  be  of  practical  value  to  the  navigator. 

109.   Azimuth  of  the  Sun  at  a  Given  Time. 

For  determining  the  error  of  the  compass  and  for  other  pur- 
poses it  is  frequently  necessary  at  sea  to  know  the  sun's  azimuth 
at  an  observed  instant  of  time.  If  the  observed  time  be  con- 
verted into  local  apparent  time  the  azimuth  Z  may  be  computed 
by  the  following  formulae.* 

tan  i  (Z+5)  =  cot  J  P  sec  J  (p  +  co-L)  cos  %(p-  co-L) ,  [112] 
taai<Z-S)  =  cot^Pcsc^(p+co-L)sm^(p-co-L).  [113] 

In  these  formulae  co-L  is  the  co-latitude.  In  practice  the  azimuth 
is  taken  from  tables  computed  by  use  of  these  formulae. 
Burdwood's  and  Davis's  Azimuth  Tables  give  the  azimuth  for 
each  degree  of  P,  L,  and  p,  the  former  ranging  from  Lat.  30°  to 
Lat.  60°  and  the  latter  from  30°  N  to  30°  S.  Publication 
No.  71  of  the  U.  S.  Hydrographic  Office  gives  azimuths  of  the 
sun  for  latitudes  up  to  61°.  For  finding  the  azimuth  of  an 
object  having  a  declination  greater  than  24°  publication  No.  120 
of  the  Hydrographic  Office  may  be  used. 

Example. 

Find  the  sun's  azimuth  when  L  =  42°  01'  N,  D  =  22°  47'  S,  P  =  gh  25™  18*. 
From  Publ.  No.  71  for  L  =  42°,  D  =  22°,  P  =  gh  20™,  the  azimuth  is  N  141°  40'  E. 
The  corresponding  azimuth  for  L  =  43°  is  141°  50',  that  is,  an  increase  of  10' 
for  i°;  the  azimuth  for  L  =  42°  D  =  23°,  and  P  =  gh  2om,  is  142°  u',  or  an 
increase  of  31'  for  i°  of  declination;  for  L  =  42°,  D  =  22°,  and  P  =  gh  30™  the 

*  Napier's  Analogies. 


NAUTICAL   ASTRONOMY  175 

azimuth  is  143?  47',  or  an  increase  of  2°  of  for  iom,  or  12'.;  for  im.  The  desired 
azimuth  is  therefore  141°  40'  +  eV  X  10'  +  f  £  X  31'  +  5-3  X  12'.;  =  143°  12'. 
The  azimuth  from  the  south  point  is  therefore  S  36°  48'  E. 

When  the  azimuth  is  determined  for  the  purpose  of  finding  the  error  of  the  com- 
pass the  observation  is  usually  taken  near  sunrise  or  sunset,  which  is  not  only  a 
convenient  time  for  making  the  pointings  at  the  sun  but  is  a  favorable  time  for  ac- 
curate determination  of  the  azimuth. 

no.   Azimuth  of  the  Sun  by  Altitude  and  Time. 

When  the  altitude  of  the  sun  is  observed  for  the  purpose  of 
finding  the  local  time,  the  azimuth  at  the  same  instant  may  be 
computed  by  the  formula 

sin  Z  =  sin  P  cos  D  sec  h.  [12] 

Example. 

Find  the  sun's  azimuth  when  P=  34°46'.4  (P.M.),  D  =  —  22°  45' 50",  h  = 
17°  4i' 

log  sin  P  =  9.  75612 
log  cos  D  =  g.  96478 
log  sec  h  =  o.  02102 

log  sin  Z    =  9.  74192 

Z    =  S33°3o'.2W 

in.   Sumner's  Method  of  Determining  a  Ship's  Position.* 

If  the  declination  of  the  sun  and  the  Greenwich  Apparent 
Time  are  known  at  any  instant,  these  two  coordinates  are  the 
latitude  and  longitude  respectively  of  a  point  on  the  earth's 
surface  which  is  vertically  under  the  sun's  centre  and  which 
may  be  called  the  "  sub-solar  "  point.  If  an  observer  were  at 
the  sub-solar  point  he  would  have  the  sun  in  his  zenith.  If 
he  were  located  i°  from  this  point,  in  any  direction,  the  sun's 
zenith  distance  would  be  i°;  if  he  were  2°  away,  the  zenith 
distance  would  be  2°.  It  is  evident,  then,  that  if  an  observer 
measures  an  altitude  of  the  sun  he  locates  himself  on  the  cir- 
cumference of  a  circle  whose  centre  is  the  sub-solar  point  and 
whose  radius  (in  degrees)  is  the  zenith  distance  of  the  sun. 
This  circle  could  be  drawn  on  a  globe  by  first  plotting  the  posi- 
tion of  the  sub-solar  point  by  means  of  its  coordinates,  and 

*  This  method  was  first  described  by  Captain  Sunmer  in  1843. 


i76 


PRACTICAL  ASTRONOMY 


then  setting  a  pair  of  dividers  to  subtend  an  arc  equal  to  the 
zenith  distance  (by  means  of  a  graduated  circle  on  the  globe) 
and  describing  a  circle  about  the  sub-solar  point  as  a  centre. 
The  observer  is  somewhere  on  this  circle  because  all  positions 
on  the  earth  where  the  sun  has  this  measured  altitude  are  located 
on  this  same  circle.  This  is  called  a  circle  of  position,  and  any 
portion  of  it  a  line  of  position  or  a  Sumner  line. 


FIG.  69 

Suppose  that  at  Greenwich  Apparent  Time  ih  the  sun  is 
observed  to  have  a  zenith  distance  of  20°,  the  declination  being 
20°  N.  The  sub-solar  point  is  then  at  A .  Fig.  69,  and  the  observer 
is  somewhere  on  the  circle  described  about  A  with  a  radius  20°. 
If  he  waits  until  the  G.  A.  T.  is  4^  and  again  observes  the  sun, 
obtaining  30°  for  his  zenith  distance,  he  locates  himself  on  the 
circle  whose  centre  is  J5,  the  coordinates  being  4h  and  (say) 
20°  02'  N,  and  the  radius  of  which  is  30°.  If  the  ship's  position 


NAUTICAL  ASTRONOMY  177 

has  not  changed  between  the  observations  it  is  either  at  5  or 
at  T;  in  practice  there  is  no  difficulty  in  deciding  which  is  the 
correct  point,  on  account  of  their  great  distance  apart.  A 
knowledge  of  the  sun's  bearing  also  shows  which  portion  of  the 
circle  contains  the  point.  If,  however,  the  ship  has  changed  its 
position  since  the  first  observation,  it  is  necessary  to  allow  for 
its  "  run "  as  follows.  Assuming  that  the  ship  has  sailed 
directly  away  from  the  sun,  say  a  distance  of  60  miles  or  i°, 
then,  if  the  first  observation  had  been  made  while  the  ship  was 
in  the  second  position,  the  point  A  would  be  the  same,  but  the 
radius  of  the  circle  would  be  21°,  locating  the  ship  on  the  dotted 
circle.  The  true  position  of  the  ship  at  the  second  observation 
is,  therefore,  at  the  intersection  S1 '.  If  the  vessel  does  not  actu- 
ally sail  directly  away  from  or  directly  toward  the  sun  it  is  a 
simple  matter  to  calculate  the  increase  or  decrease  in  radius 
due  to  the  change  in  the  observer's  zenith. 

This  is  in  principle  Sumner's  method  of  locating  a  ship. 
In  practice  the  circles  would  seldom  have  as  short  radii  as  those 
in  Fig.  69;  small  circles  were  chosen  only  for  convenience  in 
illustrating  the  method.  On  account  of  the  long  radius  of  the 
circle  of  position  only  a  small  portion  of  this  circle  can  be  shown 
on  an  ordinary  chart;  in  fact,  the  portion  which  it  is  necessary 
to  use  is  generally  so  short  that  the  curvature  is  negligible  and 
the  line  of  position  appears  on  the  chart  as  a  straight  line.  In 
order  to  plot  a  Sumner  line  on  the  chart,  two  latitudes  may  be 
assumed  between  which  the  actual  latitude  is  supposed  to  lie; 
and  from  these,  the  known  declination,  the  observed  altitude, 
and  the  chronometer  reading,  two  longitudes  may  be  computed 
(Art.  107),  one  for  each  of  the  assumed  latitudes.  This  gives 
the  coordinates  of  two  points  on  the  line  of  position  by  means 
of  which  it  may  be  plotted  on  the  chart.  Another  observation 
may  be  made  a  few  hours  later  and  the  new  line  plotted  in  a 
similar  manner.  In  order  to  allow  for  the  change  in  the  radius 
of  the  circle  due  to  the  ship's  run  between  observations,  it  is 
only  necessary  to  move  the  first  position  line  parallel  to  itself 


i78 


PRACTICAL  ASTRONOMY 


in  the  direction  of  the  ship's  course  and  a  distance  equal  to  the 
ship's  run.  In  Fig.  70,  AB  is  a  line  obtained  from  a  9  A.M 
observation  on  the  sun  and  by  assuming  the  latitudes  42°  and 
43°.  A  second  observation  is  made  at  2  P.M.;  between  gh  and 
2h  the  ship  has  sailed  S  75°  W,  67'.*  Plotting  this  run  on  the 
chart  so  as  to  move  any  point  on  AB,  such  as  x,  in  the  direction 
S  75°  W  and  a  distance  of  67',  the  new  position  line  for  the  first 


X?        43° 


42° 


FIG.  70 

observation  is  A'B' '.  The  P.M.  line  of  position  is  located  by 
assuming  the  same  latitudes,  42°  and  43°,  the  result  being  the 
line  CD.  The  point  of  intersection  S  is  the  position  of  the  ship 
at  the  time  of  the  second  observation.  Since  the  bearing  of 
the  sun  is  always  at  right  angles  to  the  bearing  of  the  Sumner 
line,  it  is  evident  that  one  point  and  the  bearing  would  be 
sufficient  to  locate  the  line  on  the  chart. 

112.   Position  by  Computation. 

The  coordinates  of  the  point  of  intersection  of  the  lines  of  position  may  be 
calculated  more  precisely  than  they  can  be  taken  from  the  chart.     When  the  first 


*  The  nautical  mile  (6080.20  feet)  is  assumed  to  be  equal  to  an  arc  of  i'  on 
any  part  of  the  earth's  surface. 


NAUTICAL  ASTRONOMY 


I79 


altitude  is  measured  the  navigator  assumes  a  latitude  which  is  near  the  true  lati- 
tude, and  from  this  calculates  the  corresponding  longitude.  The  approximate 
azimuth  of  the  sun  is  also  calculated  from  the  same  data.  (Equa.  [30].)  The 
run  of  the  ship  up  to  the  time  of  the  second  observation  is  reduced  to  the  difference 
in  latitude  and  the  difference  in  longitude  from  the  known  course  and  speed  of 
the  vessel.  These  two  differences  are  applied  as  corrections  to  the  assumed  lati- 
tude and  the  calculated  longitude.  This  places  the  ship  on  the  new  Sumner  line 
(corresponding  to  A'B',  Fig.  70).  When  the  P.M.  observation  is  made  the  corrected 
latitude  is  used  in  computing  the  new  longitude.  The  result  of  these  two  obser- 
vations is  shown  in  Fig.  71.  Point  A  is  the  first  position;  A'  is  the  position  of  A 

A 


FIG.  71 

corrected  for  the  run  of  the  ship;  B  is  the  position  obtained  by  the  P.M.  observation 
using  the  latitude  of  A'.  A'B  is  then  the  discrepancy  in  the  longitudes,  due  to 
the  fact  that  a  wrong  latitude  has  been  chosen,  and  is  the  base  of  a  triangle  the 
vertex  of  which,  C,  is  the  true  position  of  the  ship.  The  base  angles  A'  and  B 
are  the  azimuths  of  the  sun  at  the  times  of  observation.  In  practice  this  triangle 
is  often  solved  as  follows:*  Dropping  a  perpendicular  from  C  to  A'B,  forming 
two  right  triangles, 

Bd  =  Cd  cot  Z2, 
and 

A'd  =  CdcotZ1, 
or 

A/>2  =  AZ,  cot  Z2, 

A/>i  =  AL  cot  Zi, 


*  See  A.  C.  Johnson's 
Weather." 


On  Finding  the  Latitude  and  Longitude  in  Cloudy 


l8o  PRACTICAL  ASTRONOMY 

where  AL  =  the  error  in  latitude  and  Ap  the  difference  in  departure.  In  order 
to  express  Bd  and  A'd  as  differences  in  longitude  (AM)  it  is  necessary  to  introduce 
the  factor  sec  L,  giving 

AM2  =  AL  sec  L  cot  Z2,  [114! 

AM2  =  AL  sec  L  cot  Zi.  [115! 

To  find  AL,  the  correction  to  the  latitude,  the  distance  A'B  =  AM2  +  AMi 
is  known,  the  factors  sec  L  cot  Z  may  be  found  from  the  approximate  latitude 
and  the  sun's  azimuths,  therefore 

A'B 

""  sec  L  cot  Zi  +  sec  L  cot  Z2 ' 

Having  found  AL,  the  corrections  AMi  and  AM2  are  found  by  [114]  and  [115]. 
Since  the  factors  sec  L  cot  Z  are,  in  practice,  taken  from  a  table  and  the  operations 
indicated  in  Equa.  [114],  [115],  and  [116]  are  easily  performed  with  the  slide  rule 
the  method  is  in  reality  a  rapid  one. 

In  the  above  description  the  observations  are  taken  one  in  the  forenoon  and  one 
in  the  afternoon,  but  any  two  observations,  provided  the  position  lines  intersect  at  an 
angle  over  30°,  will  give  good  results.  If  the  observations  are  both  on  the  same 
side  of  the  meridian  the  denominator  of  [116]  becomes  the  difference  of  the  factors 
instead  of  the  sum.  If  two  objects  can  be  observed  at  the  same  time,  and  their 
bearings  differ  by  30°  or  more,  the  position  of  the  ship  is  obtained  at  once,  since 
there  is  no  run  of  the  ship  to  be  applied.  This  observation  might  be  made  upon 
two  bright  stars  or  planets  at  twilight.  It  should  be  observed  that  the  accuracy 
of  this  method  depends  upon  the  accuracy  of  the  chronometer,  just  as  in  the 
methods  of  Art.  107. 

One  of  the  great  advantages  of  this  method  is  that  even  if  only  one  observation 
can  be  taken  it  may  be  utilized  to  locate  the  ship  along  a  (nearly)  straight  line; 
and  this  is  often  of  great  value.  For  example,  if  the  first  position  line  is  found 
to  pass  directly  through  some  point  of  danger,  then  the  navigator  knows  the 
bearing  of  the  point,  although  he  does  not  know  his  distance  from  it;  but  with 
the  single  observation  he  is  able  to  avoid  the  danger.  In  case  it  is  a  point  which 
it  is  desired  to  reach,  the  true  course  which  the  ship  should  steer  is  at  once  known. 
The  following  example  illustrates  the  method  of  computing  the  coordinates  of  the 
point  of  intersection. 


NAUTICAL  ASTRONOMY  181 

Example. 

Location  of  ship  by  Sumner's  Method,  Jan.  4,  1910. 

At  chronometer  time  ih  i2m  48*  the  sun's  lower  limb  is  observed  to  be  15°  53' 
30":  index  corr.  =  o";  height  of  eye  =  36  ft.;  chronometer  is  15*  fast  of  G.  M.  T. 
Latitude  by  dead  reckoning,  42°  oo'  N.  At  chronometer  time  6h  05 m  46*  the  alti- 
tude of  the  sun's  lower  limb  =  17°  33'  30";  index  corr.  =  o";  height  of  eye,  36  ft; 
chronometer,  15*  fast.  The  run  between  the  observations  was  i'  N  and  60'  W. 

First  Observation 

Semidiam.  =  +  16'   17"  Decimation  at  G.  M.  N.  =  -  22°  47'    22".  3 

dip  =  -    5     53  +  i5"-i5  X  i*.2  +  18  .  z 

r&p   =  —    3     10 


Decl.  =  -  22°  47'    04".  r 
Corr.  =   +         07'  14"  p  =     112°  47'    04".  i 

Obs.  Alt.    =       15°  53'    30" 

True  Alt.  =       16°  oo'   44" 

L  =    42°  oo'       seco.  12893  Equa.  t.  =  4™  49*.  80 

p  =  112    47         esc 0.03528      is.i4S  X  iA-2  =  i  -37 

h  =    16   oo  .  7 


Cor'd  Eq.  t.  =  4TO  51*.  17 


170    47  .7 

5  =    85°  23'.  8   cos  8. 90448  Sun's  Az.*  =  S  36°  48'  E 

s—h=    69°  23  .  i    sin  9. 97126      cot  Az.  X  sec  Lat.  =  i.  80 

2)9. 03995 


log  sin  £  P  =  9.  51998 
|  P  =  19°  20'.  2 
P  =  3f  40'.  4 

=  a*     34™  41*.  6 

L.  A.  T.  =  9*     25"*  i8«.  4 

Eq.  t.  =  4     51  .  2 


L.  M.  T.  =  gh     30™  09*.  6 
G.  M.  T.  =  i       12     33 


Long.  =  3^    42m  23*.  4  Lat. 

=  55°  35'- 8  Run 

Run     =     +60' 

Cor'd  Lat.  =  42°  01' 
Cor'd  Long.  =  56°  35'.  8 

*  By  table  or  by  Equa.  [30];  see  Art.  93,  p.  155. 


182  PRACTICAL  ASTRONOMY 

Second  Observation 

semidiam.  =           +  16'  17"     Decl.  at  G.  M.  N.  =  -  22°  47'   22".  3 

dip  =           -    5  S3         +  iS"-iS  X  6A.i  =  +132.4 
r  &  p  =           —    2.  49 

Decl.  =  -22°  45'   49".  9 

Corr.  =           +7    35                                     p  =  112°  45'   49".  9 
Obs.  Alt.  =      17    33    30 

True  Alt.  =      17°  41'  05" 

L  =    42°  01'       sec  o.  12904  "Eq.  t.  =  4™  49".  80 

p  =  112    45  .8  esc 0.03522  i*.  145  X  6*.i  =           6.98 
h  =    17   41  .1 

Cor'd  Eq.  t.  =  4™  56*.  78 
172    27.9 

s  =    86°  13. 9  cos  8.  81771  Sun's  Az.*  =  S  33°  30' W 

5  —  h  =    68°  32'.  9    sin  9. 96883  cot  Az.  X  sec  Lat.  =  2. 03 


log  sin  £  P  =  9. 47540 

1  P  -   I?o    23,''  2 

L.  A.  T.    =    2h  igm  05*.  6 
Eq.t.  =  4     56  .8 

L.  M.  T.   =    2&  24™  02*.  4 
G.  M.  T.   =    6    05     31 

Long.    =    3^  41™  28s.  6 
=  55°  22'.  i 

ist  Long.  =  56°  35'.  8  19.  2  X  i.  80  =  34'.  6  Corr.  to  ist  Long. 

2d  Long.  =  55    22  .  i  19.  2  X  2. 03  =  39  .  o  Corr.  to  2d  Long. 

Diff.  =    i°  13'.  7  =  73'.  7       ist  Long.  =  56°  35'.  8    2d  Long.=  55°  22'.! 

Corr.  =         34  . 6         Corr.  =         39  -o 

i  80  4-  2 =  I9/<  2  Corr.  to  the  Lat.      Long.  =  56°  01'.  2       Long.  =  56°  oi'.i 

/.  Lat.  =  42°  20'  N.  /.  Long.  =  56°  01'  W. 

*  By  table  or  by  Equa.  [30];  see  Art.  93,  p.  155. 


TABLES 


184 


PRACTICAL  ASTRONOMY 


TABLE  I.     MEAN  REFRACTION. 
Barometer,   29.5  inches.  Thermometer,   50°  F. 


App.  Alt. 

Refr. 

App.  Alt. 

Refr. 

App.  Alt. 

Refr. 

App.  Alt. 

Refr. 

o°oo' 

33'  5i" 

10°  00' 

5'  13" 

20°  oo' 

2'  36" 

35°  oo' 

i'  21" 

30 

28  ii 

30 

4  59 

30 

2  32 

36  oo 

i  18 

I  OO 

23  51 

II  00 

4  46 

21  OO 

2  28 

37  oo 

i  16 

30 

20  33 

30 

4  34 

30 

2  24 

38  oo 

i  13 

2  00 

i7  55 

12  00 

4  22 

22  OO 

2  20 

40  oo 

i  08 

30 

15  49 

30 

4  12 

30 

2  17 

42  oo 

i  03 

7   OO 

14  07 

13  oo 

4  02 

23  oo 

2  14 

44  oo 

o  59 

30 

12  42 

30 

3  54 

30 

2  II 

46  oo 

o  55 

4  oo 

II  31 

14  oo 

3  45 

24  oo 

2  08 

48  oo 

o  5I 

30 

10  32 

30 

3  37 

30 

2  05 

50  oo 

o  48 

5  oo 

9  40 

15  oo 

3  3° 

25  oo 

2  O2 

52  oo 

o  45 

30 

8  56 

3° 

3  23 

26  oo 

i  57 

54  oo 

o  41 

6  oo 

8  19 

16  oo 

3  17 

27  oo 

52 

56  oo 

o  38 

3° 

7  45 

3° 

3  I0 

28  oo 

47 

58  oo 

o  36 

7  oo 

7  15 

17  oo 

3  °5 

29  oo 

43 

60  oo 

o  33 

3° 

6  49 

3° 

2  59 

30  oo 

39 

65  oo 

o  27 

8  oo 

6  26 

18  oo 

2  54 

31  oo 

35 

70  oo 

O  21 

30 
9  oo 

6  05 
5  46 

3° 
19  oo 

2  49 
2  44 

32  oo 
33  oo 

75  oo 
80  oo 

o  15 

0  10 

3° 

5  29 

3° 

2  40 

34  oo 

24 

85  oo 

o  05 

IO  OO 

5  J3 

20  OO 

2  36 

35  o° 

21 

90  oo 

o  oo 

TABLES 


I8S 


TABLE   II.     FOR   CONVERTING   SIDEREAL   INTO   MEAN   SOLAR 

TIME. 

(Increase  in  Sun's  Right  Ascension  for  Sidereal  h.  m.  s.) 

Mean  Time  =  Sidereal  Time  -  C. 


Sid. 
Hrs. 

Corr. 

Sid. 
Min. 

Corr. 

Sid. 
Min 

Corr. 

Sid. 
Sec. 

Corr. 

Sid. 
Sec. 

Corr. 

I 

m        s 
o     9  .  830 

I 

8 
O.l64 

31 

8 

5-°79 

I 

« 
0.003 

31 

• 
0.085 

2 

o  19.659 

2 

0.328 

32 

5.242 

2 

0.005 

32 

0.087 

3 

o  29.489 

3 

0.491 

33 

5.406 

3 

0.008 

33 

0.090 

4 

o  39.318 

4 

°-655 

34 

5-570 

4 

O.OII 

34 

0.093 

5 

o  49.  148 

5 

O.8l9 

35 

5-734 

5 

0.014 

35 

0.096 

6 

o  58.977 

6 

0.983 

36 

5.898 

6 

0.016 

36 

0.098 

7 

I    8.807 

7 

I.I47 

37 

6.062 

7 

0.019 

37 

O.IOI 

8 

I  18.636 

8 

I.3II 

38 

6.225 

8 

0.022 

38 

0.104 

9 

I  28.466 

9 

1.474 

39 

6.389 

9 

O.O25 

39 

o.  106 

10 

I  38.296 

10 

1-638 

40 

6-553 

10 

0.027 

40 

0.109 

ii 

I  48.125 

ii 

I.  802 

41 

6.717 

ii 

0.030 

4i 

0.  112 

12 

1  57-955 

12 

1.966 

42 

6.881 

12 

0.033. 

42 

0.115 

13 

2       7.784 

13 

2.130 

43 

7-045 

13 

0.035 

43 

0.117 

14 

2    17.614 

14 

2.294 

44 

7.208 

14 

0.038 

44 

0.120 

15 

2    27.443 

15 

2-457 

45 

7-372 

15 

O.O4I 

45 

0.123 

16 

2    37-273 

16 

2.621 

46 

7-536 

16 

0.044 

46 

0.126 

17 

2    47.102 

17 

2.785 

47 

7.700 

17 

0.046 

47 

O.I28 

18 

2    56.932 

18 

2.949 

48 

7.864 

18 

O.049 

48 

O.I3I 

19 

3    6.762 

19 

3-"3 

49 

8.027 

19 

0.052 

49 

0.134 

20 

3  16.591 

20 

3-277 

50 

8.191 

20 

0.055 

50 

0.137 

21 

3  26.421 

21 

3-440 

51 

8-355 

21 

0.057 

51 

0.139 

22 

3  36.250 

22 

3-604 

52 

8.519 

22 

O.O6O 

52 

O.I42 

23 

3  46.080 

23 

3.768 

53 

8.683 

23 

0.063 

53 

0.145 

24 

3  55-909 

24 

3-932 

54 

8.847 

24 

0.066 

54 

0.147 

25 

4.096 

55 

9.010 

25 

0.068 

55 

0.150 

26 

4-259 

56 

9.174 

26 

0.071 

56 

0.153 

3 

29 

4-423 
4.587 
4-751 

11 

59 

9.338 
9.502 
9.666 

11 

29 

0.074 
0.076 
0.079 

57 
58 
59 

0.156 
0.158 

0.161 

30 

4-915 

60 

9.830 

3<> 

O.O82 

60 

0.164 

1 86 


PRACTICAL  ASTRONOMY 


TABLE   III.     FOR   CONVERTING  MEAN   SOLAR   INTO   SIDEREAL 

TIME. 

(Increase  in  Sun's  Right  Ascension  for  Solar  h.  m.  s.) 
Sidereal  Time  =  Mean  Time  +  C. 


ll 

Corr. 

ll 

Corr. 

u 

Corr. 

ll 

Corr. 

c    . 

Corr. 

! 

m         s 
O     9.856 

i 

s 
0.164 

3I 

s 
5-093 

i 

a 
0.003 

31 

s 

0.085 

2 

o  19.713 

2 

0.329 

32 

5-257 

2 

0.005 

32 

0.088 

3 

o  29.569 

3 

0.493 

33 

5.421 

3 

0.008 

33 

0.090 

4 

o  39.426 

4 

0.657 

34 

5.585 

4 

O.OII 

34 

0.093 

5 

o  49.282 

5 

0.821 

35 

5-75° 

5 

0.014 

35 

0.096 

6 

o  59-139 

6 

0.986 

36 

5-914 

6 

0.016 

36 

0.099 

I 

8.995 
18.852 

8 

.150 
.314 

37 
38 

6.078 
6.242 

8 

0.019 

0.022 

37 
38 

0.  101 

o.  104 

9 

28.708 

9 

-478 

39 

6.407 

9 

O.O25 

39 

0.107 

10 

38.565 

10 

.643 

40 

6-571 

10 

O.O27 

40 

o.  no 

ii 

48.421 

ii 

.807 

41 

6-73$ 

ii 

0.030 

41 

O.II2 

12 

58.278 

12 

.971 

42 

6.900 

12 

0-033 

42 

0.115 

13 

2       8.134 

13 

2.136 

43 

7.064 

13 

0.036 

43 

0.118 

14 

2    17.991 

14 

2  .  300 

44 

7.228 

14 

0.038 

44 

0.  120 

15 

2    27.847 

15 

2.464 

45 

7-392 

15 

0.041 

45 

0.123 

16 

2    37-704 

16 

2.628 

46 

7.557 

16 

O.O44 

46 

o.  126 

ll 

2    47-560 
2    57-4I7 

17 

18 

2.793 

2-957 

47 
48 

7.721 
7-885 

17 

18 

0.047 
O.O49 

S 

o.  129 

0.131 

19 

3     7-273 

19 

3.121 

49 

8.049 

19 

O.O52 

49 

0.134 

20 

3  17-129 

20 

3-285 

50 

8.214 

20 

0-055 

50 

0.137 

21 

3  26.986 

21 

3-450 

51 

8.378 

21 

0.057 

51 

0.140 

22 

3  36.842 

22 

3.614 

52 

8.542 

22 

O.O6O 

52 

0.142 

23 

3  46.699 

23 

3-778 

53 

8.707 

23 

0.063 

53 

0.145 

24 

3  56.555 

24 

3-943 

54 

8.871 

24 

O.O66 

54 

0.148 

25 

4.107 

55 

9-035 

25 

0.068 

55 

0.151 

26 

4.271 

56 

9.199 

26 

O.O7I 

56 

0.153 

28 

4-435 
4.600 

1 

9-364 
9.528 

27 
28 

0.074 
0.077 

ll 

0.156 

o.i  60 

29 

4.764 

9.692 

29 

0.079 

59 

0.162 

30 

4.928 

60 

9-856 

30 

0.082 

60 

0.164 

TABLES 

TABLE    IV. 

PARALLAX  —  SEMIDIAMETER  —  DIP. 


I87 


(A)    Sun's  parallax. 

(C)    Dip  of  the  sea  horizon. 

Sun's  altitude. 

Sun's  parallax- 

Height  of  eye 
in  feet. 

Dip  of  sea 
horizon. 

0° 

9" 

I 

o'    59" 

10 

9 

2 

i     23 

20 

8 

3 

i     42 

3° 

8 

4 

i     58 

40 

7 

5 

2        II 

5° 

6 

6 

2       24 

60 

4 

7 

2       36 

70 

3 

8 

2       46 

80 

2 

9 

2       56 

90 

O 

10 

3     06 

ii 

12 

3     15 
3     24 

(B)     Sun's  semidiameter. 

13 

3     32 

14 

3     40 

Date. 

Semidiameter. 

15 

16 

3     48 
3     55 

17 

4       02 

Jan. 

1  6'    18" 

18 

4     09 

Feb. 

16     16 

19 

4     16 

Mar. 

16     10 

20 

4     23 

Apr. 

16    02 

21 

4     29 

May 

J5     54 

22 

4     36 

June 

15     48 

23 

4     42 

July 

15     46 

24 

4     48 

Aug. 

15     47 

25 

4     54 

Sept. 

i5     53 

26 

5     °o 

Oct. 

16    01 

27 

5     06 

Nov. 

16    09 

28 

5     ii 

Dec. 

16     15 

29 

5     X7 

3° 

5     22 

35 

5     48 

40 

6       12 

45 

6    36 

5° 

6    56 

55 

7     16 

60 

7    35 

65 

7     54 

70 

8       12 

75 

8     29 

80 

8    46 

85 

9     02 

90 

9     18 

95 

9     33 

IOO 

9     48 

i88 


PRACTICAL  ASTRONOMY 


TABLE  V.*  — LOCAL  MEAN  (ASTRONOMICAL)  TIME  OF  THE  CULMINA- 
TIONS AND  ELONGATIONS  OF  POLARIS  IN  THE  YEAR  1915 
[Computed  for  latitude  40°  north  and  longitude  90°  or  6*  west  of  Greenwich.] 


Date. 

East  elongation. 

Upper  culmi- 
nation. 

West  elonga- 
tion. 

Lower  culmi- 
nation. 

1915 
January  i  

h    m 

0  51-7 

h    m 
6  46.9 

h      m 

12  42.1 

h     m 

18  44-9 

January  15 

5  51-6 

II   46.8 

17  49.6 

February  I  
February  15 

22  45-3 

4  44-5 
3  49-2 

10  39-7 

9  44-4 

16  42.5 
IS  47-2 

March  i  

2  54.O 

8  49-2 

14  52.0 

March  15 

I   58.8 

7  54-0 

13  56.8 

April  i  

18  52  7 

0   51-9 

6  47.1 

12  49-9 

April  i? 

5  52.o 

II  54.8 

May/  

I?  57-7 
ED  ci  i 

4  49-2 

10  52.0 

May  15  

3  54-2 

9  57-0 

June  i 

2  47.6 

8  50.4 

June  15  

i  52.8 

7  55-6 

July  i  

12  55.9 

18  51.1 

o  50.2 

6  53-0 

July  15. 

5  58.2 

August  i  

10   54   * 

4  51-7 

August  15 

3  56-9 

September  i  

e-j    2 

2  50-3 

September  15 

7  (ft  1 

I  55-4 

October  i  

7  50.  6 
6  55  5 

12  50  7 

18  45  9 

o  52  7 

October  15  
November  I  .  .  .  . 

6  oo.  6 

ii  55-8 

17  Si.o 

23  53-8 

22  46  9 

November  15 

v  B  a 

December  i  . 

8  50  8 

20  48  8 

December  15  

2  00.4 

7  55-6 

13  50.8 

19  S3.  6 

A.   To  refer  the  above  tabular  quantities  to  years  other  than  1915. 


For  year 

1916 

(add 
\  subtract 

1.6 
2-3 

up  to  March  i 
on  and  after  March  i 

1917 

subtract 

0.7 

1918 

add 

0.9 

1919 

add 

2-5 

i  add 

4.0 

up  to  March 

i 

1920 

)  add 

O.I 

on  and  after 

March 

i 

1921 

add 

1.6 

1922 

add 

3.1 

1923 

add 

4-5 

1924 

jadd 
jadd 

5.9  up  to  March 
2.0  on  and  after 

March 

i 

1925 

add 

3-3 

1926 

add 

4.6 

1927 

add 

5-9 

B.   To  refer  to  any  calendar  day  other  than  the  first  and  fifteenth  of  each  month 
SUBTRACT  the  quantities  below  from  the  tabular  quantity  for  the  PRECEDING  DATE. 


Day  of 
month. 

Minutes. 

No.  of  days 
elapsed. 

Day  of 
month. 

Minutes. 

No.  of  days 
elapsed. 

2or  16 

I  ;i 
i  % 

7       21 
8       22 

9      23 

3-9 
7-8 
II.  8 
IS-7 
19.6 
23-S 
27-4 
31-4 

i 

2 
3 
4 
5 
6 
7 
8 

10  or  24 

II          25 
12          26 

13        27 
14       28 

29 

30 
31 

35.3 

39-2 
43.1 
47-0 
Si.o 
54-9 
58.8 
62.7 

9 

10 

ii 

12 
13 
14 

3 

*  Furnished  by  U.  S.  Coast  and  Geodetic  Survey. 


TABLES 


189 


C.  To  refer  the  table  to  Standard  time  and  to  the  civil  or  common  method  of 
reckoning: 

(a)  Add  to  the  tabular  quantities  four  minutes  for  every  degree  of  longitude 
the  place  is  west  of  the  Standard  meridian  and  SUBTRACT  when  the  place  is  east 
of  the  Standard  meridian. 

(&)  The  astronomical  day  begins  twelve  hours  after  the  civil  day,  i.e.,  begins 
at  noon  on  the  civil  day  of  the  same  date,  and  is  reckoned  from  o  to  24  hours. 
Consequently  an  astronomical  time  less  than  twelve  hours  refers  to  the  same  civil 
day,  whereas  an  astronomical  time  greater  than  twelve  hours  refers  to  the  morn- 
ing of  the  next  civil  day. 

D.  To  refer  to  any  other  than  the  tabular  latitude  between  the  limits  of  10°  and  50° 
north:  ADD  to  the  time  of  west  elongation  om.io  for  every  degree  south  of  40°  and 
SUBTRACT  from  the  time  of  west  elongation  om.i6  for  every  degree  north  of  40°. 
Reverse  these  operations  for  correcting  times  of  east  elongation. 

E.  To  refer  to  any  other  than  the  tabular  longitude:  ADD  om.i6  for  each  15°  east 
of  the  ninetieth  meridian  and  SUBTRACT  om.i6  for  each  15°  west  of  the  ninetieth 
meridian. 

TABLE  VI.     CORRECTION  TO  THE  ALTITUDE  OF  POLARIS  * 

(Equa.  [go],  Art.  69.) 


Latitudes. 

H.A. 

10° 

15° 

20° 

25° 

30° 

35° 

40° 

45° 

50° 

ss° 

_o 

// 

IO 
20 

I 

O 

I 

2 

2 

3 

4 

4 

5 

6 

7 

3° 

2 

3 

4 

5 

6 

8 

9 

n 

13 

16 

40 

3 

5 

7 

9 

n 

J3 

IS 

18 

22 

26 

50 

5 

7 

IO 

12 

15 

18 

22 

26 

31 

37 

60 
70 

6 
7 

9 

IO 

12 

14 

15 

18 

19 

22 

23 
27 

27 
32 

II 

39 
46 

47 

80 

7 

n 

15 

20 

24 

29 

35 

42 

49  , 

60 

90 

8 

n 

16 

20 

25 

30 

36 

43 

51 

61 

100 

7 

n 

15 

19 

24 

29 

35 

41 

49 

59 

no 

6 

10 

13 

17 

21 

26 

37 

44 

53 

120 

5 

8 

n 

15 

18 

22 

26 

37 

45 

130 

4 

6 

9 

n 

14 

17 

20 

24 

29 

35 

140 

3 

4 

6 

8 

10 

12 

14 

i7 

20 

24 

ISO 

2 

3 

4 

5 

6 

7 

9 

IO 

12 

i5 

160 

I 

i 

2 

2 

3 

3 

4 

5 

6 

7 

170 

0 

0 

O 

I 

i 

i 

i 

i 

2 

2 

*  This  table  is  calculated  for  a  polar  distance  =  i°  10'.  An  increase  of  i'  in  the 
polar  distance  produces  an  increase  of  about  3%  in  the  tabulated  term.  The  hour 
angle  in  the  table  is  measured  from  o°  at  upper  culmination  either  to  the  east  or  to 
the  west. 


PRACTICAL  ASTRONOMY 


TABLE  VII. 
VALUES  OF  FACTOR  112.5  X  3600  X  SIN  i"  TAN  Ze. 


Ze 

Factor. 

Ze 

Factor. 

Ze 

Factor. 

i°oo' 

•°343 

I°20' 

•°457 

i°4o' 

•0571 

01 

.0348 

21 

.0463 

4i 

•°577 

02 

.0354 

22 

.0468 

42 

•0583 

03 

.0360 

23 

.0474 

43 

.0589 

04 

.0366 

24 

.0480 

44 

•  0594 

05 

.0371 

25 

.0486 

45 

.0600 

06 

•°377 

26 

.0491 

46 

.0606 

11 

•0383 
.0388 

27 
28 

.0497 
•°5°3 

47 

48 

.0611 
.0617 

09 

.0394 

29 

.0508 

49 

.0623 

10 

.0400 

3° 

.0514 

5° 

.0629 

II 

.0406 

3i 

.0520 

51 

.0634 

12 

.0411 

32 

.0526 

52 

.0640 

13 

.0417 

33 

•o53i 

53 

.0646 

14 

.0423 

34 

•°537 

54 

.0651 

15 

.0428 

35 

•°543 

55 

.0657 

16 

•  0434 

36 

.0548 

56 

.0663 

17 

.0440 

37 

•0554 

57 

.0669 

18 

.0446 

38 

.0560 

58 

.0674 

19 

.0451 

39 

.0566 

59 

.0680 

Letters. 

A,  a, 


A,  «, 


I,', 
K,*, 


GREEK   ALPHABET 

Name.  Letters.  Name. 

Alpha  N,  i>,  Nu 

Beta  H,  f,  Xi 

Gamma  O,  o,  Omicron 

Delta  n,  TT,  Pi 

Epsilon  P,  p,  Rho 

Zeta  S,  0",  5,  Sigma 

Eta  T,  r,  Tau 

Theta  T,  v,  Upsilon 

Iota  O,  <t>,  Phi 

Kappa  X,  x,  Chi 

Lambda  ¥,  ^,  Psi 

Mu  O,  a,,  Omega 


TABLES  igi 

ABBREVIATIONS   USED   IN   THIS   BOOK 

T  or  V  =  vernal  equinox. 
R.  A.  or  Rt.  Asc.  =  right  ascension. 
D  or  Decl.  =  decimation. 

p  =  polar  distance. 
h  or  Alt.  =  altitude. 

z  =  zenith  distance. 
P  or  H.  A.  =  hour  angle. 
L  or  Lat.  =  latitude. 
Long.  =  longitude. 
Sid.  =  sidereal. 
Sol.  =  solar. 

G.  M.  N.  =  Greenwich  Mean  Noon. 
G.  M.  T.  =  Greenwich  Mean  Time. 
G.  A.  T.  =  Greenwich  Apparent  Time. 
G.  S.  T.  =  Greenwich  Sidereal  Time. 
L.  M.  N.  =  Local  Mean  Noon. 
L.  M.  T.  =  Local  Mean  Time. 
L.  A.  T.  =  Local  Apparent  Time. 
L.  S.  T.  =  Local  Sidereal  Time. 
Eq.  T.  =  equation  of  time. 
Astr.  =  astronomical  time. 
Civ.  ==  civil. 

E.  S.  T.  =  Eastern  Standard  Time. 
U.  C.  =  upper  culmination. 
L.  C.  =  lower  culmination. 
©  or  U.  L.  =  upper  limb. 
©  or  L.  L.  =  lower  limb. 

RL  ,LL  =  right  limb,  left  limb. 

=  star. 

Corr.  =  correction. 
I.  C.  =  index  correction. 
r.  or  refr.  =  refraction  correction. 

p  =  parallax  correction. 
s.  d.  =  semidiameter. 
Z  or  Az.  =  azimuth. 
N,  E,  S,  W  =  north,  east,  south,  west. 


APPENDIX 

THE  TIDES 
The  Tides. 

The  engineer  may  occasionally  be  called  upon  to  determine 
the  height  of  mean  sea  level  or  of  mean  low  water  as  a  datum 
for  levelling  or  for  soundings.  The  exact  determination  of  these 
heights  requires  a  long  series  of  observations,  but  an  approxi- 
mate determination,  sufficiently  accurate  for  many  purposes, 
may  be  made  by  means  of  a  few  observations.  In  order  to 
make  these  observations  in  such  a  way  as  to  secure  the  best 
results  the  engineer  should  understand  the  general  theory  of 
the  tides. 

Definitions. 

The  periodic  rise  and  fall  of  the  surface  of  the  ocean,  caused 
by  the  moon's  and  the  sun's  attraction,  is  called  the  tide.  The 
word  "  tide  "  is  sometimes  applied  to  the  horizontal  movement 
of  the  water  (tidal  currents),  but  in  the  following  discussion 
it  will  be  used  only  to  designate  the  vertical  movement.  When 
the  water  is  rising  it  is  called  flood  tide;  when  it  is  falling  it  is 
called  ebb  tide.  The  maximum  height  is  called  high  water;  the 
minimum  is  called  low  water.  The  difference  between  the  two 
is  called  the  range  of  tide. 

Cause  of  the  Tides. 

The  principal  cause  of  the  tide  is  the  difference  in  the  force 
of  attraction  exerted  by  the  moon  upon  different  parts  of  the 
earth.  Since  the  force  of  attraction  varies  inversely  as  the 
square  of  the  distance,  the  portion  of  the  earth's  surface  nearest 
the  moon  is  attracted  with  a  greater  force  than  the  central 
portion,  and  the  latter  is  attracted  more  powerfully  than  the 
portion  farthest  from  the  moon.  If  the  earth  and  moon  were 
at  rest  the  surface  of  the  water  beneath  the  moon  would  be 

192 


THE  TIDES  193 

elevated  as  shown  in  Fig.  72  at  A.  And  since  the  attraction 
at  B  is  the  least,  the  water  surface  will  also  be  elevated  at  this 
point.  The  same  forces  which  tend  to  elevate  the  surface  at 
A  and  B  tend  to  depress  it  at  C  and  D.  If  the  earth  were 
set  rotating,  an  observer  at  any  point  0,  Fig.  72,  would  be 
carried  through  two  high  and  two  low  tides  each  day,  the  approx- 
imate interval  between  the  high  and  the  low  tides  being  about 
6J  hours.  This  explanation  shows  what  would  happen  if 
the  tide  were  developed  while  the  two  bodies  were  at  rest;  but, 
owing  to  the  high  velocity  of  the  earth's  rotation,  the  shallow- 
ness  of  the  water,  and  the  interference  of  continents,  the  actual 


Moon 


tide  is  very  complex.  If  the  earth's  surface  were  covered  with 
water,  and  the  earth  were  at  rest,  the  water  surface  at  high 
tide  would  be  about  two  feet  above  the  surface  at  low  tide. 
The  interference  of  continents,  however,  sometimes  forces  the 
tidal  wave  into  a  narrow,  or  shallow,  channel,  producing  a 
range  of  tide  of  fifty  feet  or  more,  as  in  the  Bay  of  Fundy. 

The  sun's  attraction  also  produces  a  tide  like  the  moon's, 
but  considerably  smaller.  The  sun's  mass  is  much  greater 
than  the  moon's  but  on  account  of  its  greater  distance  the  ratio 
of  the  tide-producing  forces  is  only  about  2  to  5.  The  tide 
actually  observed,  then,  is  a  combination  of  the  sun's  and  the 
moon's  tides. 


194 


PRACTICAL  ASTRONOMY 


Effect  of  the  Moon's  Phase. 

When  the  moon  and  the  sun  are  acting  along  the  same  line,  at 
new  or  full  moon,  the  tides  are  higher  than  usual  and  are  called 
spring  tides.  When  the  moon  is  at  quadrature  (first  or  last  quar- 
ter), the  sun's  and  the  moon's  tides  partially  neutralize  each  other 
and  the  range  of  tide  is  less  than  usual;  these  are  called  neap  tides. 

Effect  of  Change  in  Moon's  Declination. 

When  the  moon  is  on  the  equator  the  two  successive  high 
tides  are  of  nearly  the  same  height.  When  the  moon  is  north 


FIG.  73 

or  south  of  the  equator  the  two  differ  in  height,  as  is  shown  in 
Fig.  73.  At  point  B  under  the  moon  it  is  high  water,  and  the 
depth  is  greater  than  the  average.  At  Bf,  where  it  will  again 
be  high  water  about  i2h  later,  the  depth  is  less  than  the  average. 
This  is  known  as  the  diurnal  inequality.  At  the  points  E  and  Q, 
on  the  equator,  the  two  tides  are  equal. 

Effect  of  the  Moon's  Change  in  Distance. 

On  account  of  the  large  eccentricity  of  the  moon's  orbit 
the  tide-raising  force  varies  considerably  during  the  month. 
The  actual  distance  of  the  moon  varies  about  13  per  cent,  and 
as  a  result  the  tides  are  about  20  per  cent  greater  when  the  moon 
is  nearest  the  earth,  at  perigee,  than  they  are  when  the  moon 
is  farthest,  at  apogee. 


THE  TIDES  195 

Priming  and  Lagging  of  the  Tides. 

On  the  days  of  new  and  full  moon  the  high  tide  at  any  place 
follows  the  moon's  meridian  passage  by  a  certain  interval  of 
time,  depending  upon  the  place,  which  is  called  the  establish- 
ment of  the  port.  For  a  few  days  after  new  or  full  moon  the 
crest  of  the  combined  tidal  wave  is  west  of  the  moon's  tide  and 
high  water  occurs  earlier  than  usual.  This  is  called  the  priming 
of  the  tide.  For  a  few  days  before  new  or  full  moon  the  crest 
is  east  of  the  moon's  tide  and  the  time  of  high  water  is  delayed. 
This  is  called  lagging  of  the  tide. 

All  of  these  variations  are  shown  in  Fig.  74,  which  was  con- 
structed by  plotting  the  predicted  times  and  heights  from  the  U.  S. 
Coast  Survey  Tide  Tables  and  joining  these  points  by  straight 
lines.  It  will  be  seen  that  at  the  time  of  new  and  full  moon  the 
range  of  tide  is  greater  than  at  the  first  and  last  quarters;  at  the 
points  where  the  moon  is  farthest  north  or  south  of  the  equator 
(shown  by  N,  S,)  the  diurnal  inequality  is  quite  marked, 
whereas  at  the  points  where  the  moon  is  on  the  equator  (£) 
there  is  no  inequality;  at  perigee  (P)  the  range  is  much  greater 
than  at  apogee  (A). 

Effect  of  Wind  and  Atmospheric  Pressure. 

The  actual  height  and  time  of  a  high  tide  may  differ  consider- 
ably from  the  normal  values  at  any  place,  owing  to  the  weather 
conditions.  If  the  barometric  pressure  is  great  the  surface  is 
depressed,  and  vice  versa.  When  the  wind  blows  steadily  into 
a  bay  or  harbor  the  water  is  piled  up  and  the  height  of  the  tide 
is  increased.  The  time  of  high  water  is  delayed  because  the 
water  continues  to  flow  in  after  the  true  time  of  high  water  has 
passed;  the  maximum  does  not  occur  until  the  ebb  and  the  effect 
of  wind  are  balanced. 

Observation  of  the  Tides. 

In  order  to  determine  the  elevation  of  mean  sea  level,  or, 
more  properly  speaking,  of  mean  half-tide,  it  is  only  necessary 
to  observe,  by  means  of  a  graduated  staff,  the  height  of  high 
and  low  water  for  a  number  of  days,  the  number  depending  upon 


196 


PRACTICAL  ASTRONOMY 


THE  TIDES  197 

the  accuracy  desired,  and  to  take  the  mean  of  the  gauge  read- 
ings. If  the  height  of  the  zero  point  of  the  scale  is  referred  to 
some  bench  mark,  by  means  of  a  line  of  levels,  the  height  of  the 
bench  mark  above  mean  sea  level  may  be  computed.  In  order 
to  take  into  account  all  of  the  small  variations  in  the  tides 
it  would  be  necessary  to  carry  on  the  observations  for  a  series 
of  years;  a  very  fair  approximation  may  be  obtained,  however, 
in  one  lunar  month,  and  a  rough  result,  close  enough  for  many 
purposes,  may  be  obtained  in  a  few  days. 

Tide  Gauges. 

If  an  elaborate  series  of  observations  is  to  be  made,  the  self- 
registering  tide  gauge  is  the  best  one  to  use.  This  consists  of 
a  float,  which  is  enclosed  in  a  vertical  wooden  box  and  which 
rises  and  falls  with  the  tide.  A  cord  is  attached  to  the  float 
and  is  connected  by  means  of  a  reducing  mechanism  with  the 
pen  of  a  recording  apparatus.  The  record  sheet  is  wrapped 
about  a  cylinder,  which  is  revolved  by  means  of  clockwork. 
As  the  tide  rises  and  falls  the  float  rises  and  falls  in  the  box 
and  the  pen  traces  out  the  tide  curve  on  a  reduced  scale.  The 
scale  of  heights  is  found  by  taking  occasional  readings  on  a 
staff  gauge  which  is  set  up  near  the  float  box  and  referred  to  a 
permanent  bench  mark.  The  time  scale  is  found  by  means  of 
reference  marks  made  on  the  sheet  at  known  times. 

When  only  a  few  observations  are  to  be  made  the  staff  gauge 
is  the  simplest  to  construct  and  to  use.  It  consists  of  a  vertical 
graduated  staff  fastened  securely  in  place,  and  at  such  a  height 
that  the  elevation  of  the  water  surface  may  be  read  on  the 
graduated  scale  at  any  time.  Where  the  water  is  compara- 
tively still  the  height  may  be  read  directly  on  the  scale;  but 
where  there  are  currents  or  waves  the  construction  must  be 
modified.  If  a  current  is  running  rapidly  by  the  gauge  but 
the  surface  does  not  fluctuate  rapidly,  the  ripple  caused  by  the 
water  striking  the  gauge  may  be  avoided  by  fastening  wooden 
strips  on  the  sides  so  as  to  deflect  the  current  at  a  slight 
angle.  The  horizontal  cross  section  of  such  a  gauge  is  shown  in 


1 98 


PRACTICAL  ASTRONOMY 


Fig.  75.     If  there  are  waves  on  the  surface  of  the  water  the  height 
will  vary  so  rapidly  that  accurate  readings  cannot  be  made.     In 

order  to  avoid  this  difficulty  a 
glass  tube  about  f  inch  in  di- 
ameter is  placed  between  two 
wooden  strips  (Fig.  76),  one  of 
which  is  used  for  the  graduated 


FIG.  75 


scale.  The  water  enters  the  glass  tube  and  stands  at  the  height 
of  the  water  surface  outside.  In  order  to*  check  sudden  varia- 
tions in  height  the  water  is  allowed  to  enter  this  tube  only 
through  a  very  small  tube  (imm  inside  diameter)  placed  in  a 
cork  or  rubber  stopper  at  the  lower  end 
of  the  large  tube.  The  water  can  rise 
in  the  tube  rapidly  enough  to  show  the 
general  level  of  the  water  surface,  but 
small  waves  have  practically  no  effect 
upon  the  reading.  For  convenience  the 
gauge  is  made  in  sections  about  three 
feet  long.  These  may  be  placed  end  to 
end  and  the  large  tubes  connected  by 
means  of  the  smaller  ones  passing 
through  the  stoppers.  In  order  to  read 
the  gauge  at  a  distance  it  is  convenient 
to  have  a  narrow  strip  of  red  painted 
on  the  back  of  the  tube  or  else  blown 
into  the  glass.*  Above  the  water  surface 
this  strip  shows  its  true  size,  but  below 
the  surface,  owing  to  the  refraction  of 
light  by  the  water,  the  strip  appears 
several  times  its  true  width,  making 
it  easy  to  distinguish  the  dividing  line. 
Such  a  gauge  may  be  read  from  a  considerable  distance  by 
means  of  a  transit  telescope  or  field  glasses. 


FIG.  76 


Tubes  of  this  sort  are  manufactured  for  use  in  water  gauges  of  steam  boilers. 


THE  TIDES  199 

Location  of  Gauge. 

The  spot  chosen  for  setting  up  the  gauge  should  be  near  the 
open  sea,  where  the  true  range  of  tide  will  be  obtained.  It 
should  be  somewhat  sheltered,  if  possible,  against  heavy  seas. 
The  depth  of  the  water  and  the  position  of  the  gauge  should  be 
such  that  even  at  extremely  low  or  extremely  high  tides  the 
water  will  stand  at  some  height  on  the  scale. 

Making  the  Observations. 

The  maximum  and  minimum  scale  readings  at  the  times  of 
high  and  low  tides  should  be  observed,  together  with  the  times 
at  which  they  occur.  The  observations  of  scale  readings  should 
be  begun  some  thirty  minutes  before  the  predicted  time  of  high 
or  low  water,  and  continued,  at  intervals  of  about  5 w,  until  a 
little  while  after  the  maximum  or  minimum  is  reached.  The 
height  of  the  water  surface  sometimes  fluctuates  at  the  time 
of  high  or  low  tide,  so  that  the  first  maximum  or  minimum 
reached  may  not  be  the  true  time  of  high  or  low  water.  In 
order  to  determine  whether  the  tides  are  normal  the  force  and 
direction  of  the  wind  and  the  barometric  pressure  may  be 
noted. 

Reducing  the  Observations. 

If  the  gauge  readings  vary  so  that  it  is  difficult  to  determine 
by  inspection  where  the  maximum  or  minimum  occurred,  the 
observations  may  be  plotted,  taking  the  times  as  abscissae  and 
gauge  readings  as  ordinates.  A  smooth  curve  drawn  through 
the  points  so  as  to  eliminate  accidental  errors  will  show  the  posi- 
tion of  the  maximum  or  minimum  point.  (Figs.  77a  and  77b.) 
When  all  of  the  observations  have  been  worked  up  in  this  way 
the  mean  of  all  of  the  high-water  and  low-water  readings  may 
be  taken  as  the  scale  reading  for  mean  half- tide.  There  should 
of  course  be  as  many  high-water  readings  as  low-water  readings. 
If  the  mean  half-tide  must  be  determined  from  a  very  limited 
number  of  observations,  these  should  be  combined  in  pairs 
in  such  a  way  that  the  diurnal  inequality  does  not  introduce 
an  error.  In  Fig.  78  it  will  be  seen  that  the  mean  of  a  and  b, 


2OO 


PRACTICAL  ASTRONOMY 


or  the  mean  of  c  and  d,  or  e  and/,  will  give  nearly  the  mean  half- 
tide;  but  if  b  and  c,  or  d  and  e,  are  combined,  the  mean  is  in 


HIGH   WATER 

MACHIAS   BAY,    ME. 

JUNE   8,   1905. 


146 


14.5  <i*CL 


Eastern  Time 
FIG.  77a 

Eastern  Time 


LOW  WATER 

MACHIAS  BAY,   ME. 

JUNE   10.    1905. 


1.90 


FIG.  7;b 

one  case  too  small  and  in  the  other  case  too  great.  The  proper 
selection  of  tides  may  be  made  by  examining  the  predicted 
heights  and  times  given  in  the  tables  issued  by  the  U.  S.  Coast 


THE   TIDES  201 

and  Geodetic  Survey.  By  examining  the  predicted  heights  the 
exact  relation  may  be  found  between  mean  sea  level  and  the 
mean  half-tide  as  computed  from  the  predicted  heights  corre- 
sponding to  those  tides  actually  observed.  The  difference  be- 
tween these  two  may  be  applied  as  a  correction  to  the  mean 
of  the  observed  tides  to  obtain  mean  sea  level.  For  example, 
suppose  that  the  predicted  heights  at  a  port  near  the  place  of 
observation  indicate  that  the  mean  of  a,  b,  c,  d,  e,  and/  is  0.2  ft. 


FIG.  78 

below  mean  sea  level.  Then  if  these  six  tides  are  observed  and 
the  results  averaged,  a  correction  of  0.2  ft.  should  be  added  to 
the  mean  of  the  six  heights  in  order  to  obtain  mean  sea  level. 

Prediction  of  Tides. 

Since  the  local  conditions  have  such  a  great  influence  in 
determining  the  tides  at  any  one  place,  the  prediction  of  the 
times  and  heights  of  high  and  low  water  for  that  place  must  be 
based  upon  a  long  series  of  observations  made  at  the  same  point. 
Tide  Tables  giving  predicted  tides  for  one  year  are  published 


202  PRACTICAL  ASTRONOMY 

annually  by  the  United  States  Coast  and  Geodetic  Survey; 
these  tables  give  the  times  and  heights  of  high  and  low  water 
for  the  principal  ports  of  the  United  States,  and  also  for  many 
foreign  ports.  The  method  of  using  these  tables  is  explained 
in  a  note  at  the  foot  of  each  page.  A  brief  statement  of  the 
theory  of  tides  is  given  in  the  Introduction. 

The  approximate  time  of  high  water  at  any  place  may  be 
computed  from  the  time  of  the  moon's  meridian  passage,  pro- 
vided we  know  the  average  interval  between  the  moon's  transit 
and  the  following  high  water,  i.e.,  the  "  establishment  of  the 
port."  The  mean  time  of  the  moon's  transit  over  the  meridian 
of  Greenwich  is  given  in  the  Nautical  Almanac  for  each  day, 
together  with  the  change  per  hour  of  longitude.  The  local 
time  of  transit  is  computed  by  adding  to  the  tabular  time  the 
hourly  change  multiplied  by  the  number  of  hours  in  the  west 
longitude;  this  result,  added  to  the  establishment  of  the  port, 
gives  the  approximate  time  of  high  water.  The  result  is  nearly 
correct  at  the  times  of  new  and  full  moon,  but  at  other  times 
is  subject  to  a  few  minutes  variation. 


INDEX 


Aberration  of  light,  12 
Adjustment  of  transit,  83,  87 
Almucantar,  15,  83,  in 
Altitude,  19 

of  pole,  27 

Angle  of  the  vertical,  73 
Annual  aberration,  13 
Aphelion,  9 
Apparent  motion,  3,  28 

time,  41 
Arctic  circle,  30 
Aries,  first  point  of,  16 
Astronomical  time,  44 

transit,  87,  117 

triangle,  31,  120 
Atlantic  time,  56 
Attachments  to  transit,  86 
Autumnal  equinox,  16 
Axis,  3,  8 
Azimuth,  19,  146 

mark,  146 

tables,  174 

Bearings,  19 
Besselian  year,  68 

Calendar,  59 

Celestial  latitude  and  longitude,  22 

sphere,  i 
Central  time,  56 
Chronograph,  93,  136,  141 
Chronometer,  92,  141,  173 

correction,  114 

Circumpolar  star,  29,  103,  155 
Civil  time,  44 
Co-latitude,  22 
Colure,  17 
Comparison  of  chronometer,  93 


Constant  of  aberration,  13 
Constellations,  10,  98 
Cross  hairs,  82,  87 
Culmination,  39,  61,  103 
Curvature,  120,  153,  158 

Date  line,  58 

Dead  reckoning,  172 

Declination,  20 

parallels  of,  16 
Dip,  79 
Diurnal  aberration,  13,  158 

inequality,  194 

Eastern  time,  56 
Ebb  tide,  192 
Ecliptic,  16,  100,  102 
Elongation,  36,  147 
Ephemeris,  62 

Equal  altitude  method,  128,  164 
Equation  of  time,  41 
Equator,  15 

systems,  19 
Equinoxes,  9,  16 
Errors  in  horizontal  angle,  97 

in  transit  observations,  88,  118 
Eye  and  ear  method,  96 
Eyepiece,  prismatic,  87,  118 

Figure  of  the  earth,  72 
Fixed  stars,  2,  4,  68 
Flood  tide,  192 
Focus,  104 

Gravity,  82 
Gravitation,  7 
Greenwich,  23,  45,  52,  172 
Gyroscope,  12 


203 


204 


INDEX 


Hemisphere,  9 
Horizon,  14 

artificial,  91 

system,  19,  83 
Hour  angle,  20,  36 

circle,  16 
Hydrographic  office,  134,  174 

Index  error,  84,  90,  106 
Interpolation,  69 

Lagging,  195 
Latitude,  22,  27,  103 

astronomical,  geocentric  and  geodetic, 
72 

at  sea,  170 

reduction  of,  73 
Leap-year,  59 
Level  correction,  158 
Local  time,  45 
Longitude,  22,  45,  139 

at  sea,  172 

Lunar  distance,  172 

Magnitudes,  99 
Mean  sun,  41,  55 

time,  41 
Meridian,  16 
Micrometer,  94,  112,  156 
Midnight  sun,  30 
Moon,  apparent  motion  of,  5 

culminations,  69,  141 
Motion,  apparent,  3,  28 
Mountain  time,  56 

Nadir,  14 

Nautical  almanac,  43,  62 

mile,  178 
Neap  tide,  194 
Nutation,  10 

Object  glass,  82,  87 
Obliquity  of  ecliptic,  8,  n,  16 


Observations,  62 
Observing,  95 

Observer,  coordinates  of,  22 
Orbit,  3 
of  earth,  7 

Pacific  time,  56 

Parallactic  angle,  31,  134,  154 

Parallax,  63,  73 

correction,  74 

horizontal,  75 
Parallel  of  altitude,  15 

of  declination,  16 

sphere,  29 
Perihelion,  9 

Phases  of  the  moon,  144,  194 
Planets,  3,  102,  135 
Plumb-line,  14,  72 
Pointers,  100 
Pole,  3,  n,  15 

star,  99,  162 
Polar  distance,  20 
Precession,  10,  101 
Prediction  of  tides,  201 
Primary  circle,  18 
Prime  vertical,  16,  122,  172 
Priming,  195 
Prismatic  eyepiece,  87,  118,  152 

Radius  vector,  41 
Range,  135 

of  tide,  192 
Rate,  114 
Reduction  to  elongation,  150 

of  latitude,  73 

to  the  meridian,  109 
Refraction,  76 

correction,  76 

effect  on  dip,  80 

index  of,  77 
Retrograde  motion,  6 
Right  ascension,  20,  36 

sphere,  28 
Rotation,  3,  39 
Run  of  ship,  177 


INDEX 


205 


Sea-horizon,  170 
Seasons,  7 

Secondary  circles,  18 
Semidiameter,  63,  78 

contraction  of,  79 
Sextant,  80,  88,  170 
Sidereal  day,  39 

time,  40,  49,  52 
Signs  of  the  Zodiac,  100 
Solar  day,  40 

time,  40,  49,  52 

system,  2 
Solstice,  1 6 

Spherical  coordinates,  18,  31 
Spheroid,  10,  72 
Spirit  level,  14 
Spring  tides,  194 
Stadia  hairs,  151 
Standard  time,  56 
Standards  of  transit,  82 
Star  catalogues,  69,  94 

fixed, 4 

list,  119,  130 

nearest,  2 

Striding  level,  86,  87,  115,  156 
Sub-solar  point,  175 
Summer,  9 

Sumner's  method,  175 
Sumner  line,  176,  179 
Sun,  altitude  of,  105 

apparent  motion  of,  5 

dial,  41 

fictitious,  41 

glass,  87 


Talcott's  method,  69,  94,  112 
Telegraph  method,  140 

signals,  136 
Tides,  192 
Tide  gauge,  197 

tables,  201 
Time  ball,  137 

service,  136 

sight,  172 

star,  125 
Transit,  astronomical,  87 

engineer's,  78,  82 

time  of,  39 

Transportation  of  timepiece,  139 
Tropical  year,  50 

Vernal  equinox,  16 
Vernier  of  sextant,  90 

of  transit,  82 
Vertical  circle,  14,  124 

line,  14 
Visible  horizon,  14 

Washington,  62 

Watch  correction,  114,  139 

Winter,  8 

Wireless  telegraph  signals,  137 

Year,  50,  68 

Zenith,  14 

distance,  19 

telescope,  94,  112 
Zodiac,  100 


43 5853 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


